European Mathematical Society - 11 Number theory
https://euro-math-soc.eu/msc/11-number-theory
enClosing the Gap
https://euro-math-soc.eu/review/closing-gap
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Vicky Neale has a degree in number theory and is now lecturer at the Balliol College, University of Oxford. She has a reputation to be an excellent communicator. This also shows in this marvellous booklet in which she gives a general introduction to the advances made in the period 2013-2014 in the quest for a solution of the twin prime conjecture. But she also explains how mathematicians think and collaborate.</p>
<p>The twin prime conjecture is claiming that there are infinitely many prime numbers whose difference is 2 like 3 and 5 or 11 and 13. It is easy to explain what prime numbers are, and it is even possible for anyone to understand Euclid's proof that there are infinitely many primes. The twin prime conjecture is however still one of the long standing open unsolved problems: easy to formulate and understand but hard to solve. Several attempts and generalizations were formulated. For example it can be claimed there are infinitely many primes whose difference is an even positive integer N. The twin prime conjecture corresponds to N = 2.</p>
<p>And then, in April 2013, Yitang Zhang could prove that the latter generalization holds for N equal to 70.000.000, a major breakthrough. Within a year N was reduced to 246. Neale presents the different steps that were obtained in this reduction almost month by month as a thrilling adventurous quest.</p>
<p>Scott Morrison and Terence Tao, two mathematical bloggers quickly used Zhang's approach to reduce the N to 42.342.946. Tim Gowers, another active blogger proposed a massive collaboration and a Polymath project was set up by Tao. This Polymath platform is a totally new way of collaboration between mathematicians that Gowers had proposed back in 2009. The blog is fully in the open and anyone who wants to take part can dump some guesses or partial ideas on the website. The results are published under the author name D.H.J. Polymath and the website shows who has collaborated in the discussion. Neale spends some pages to discuss this kind of collaboration and comments on its advantages and disadvantages. The project on the twin primes was numbered Polymanth8 and it turned out to be particularly successful. The problem that had been out for so long now progressed quickly because already in June 2013, N was down to 12.006. In July they reached 4.689.</p>
<p>But while in August 2013 Tao is announces to write up the paper with the Polymath8 result, another twist of plot occurs. James Maynard posted a paper on arXiv in November 2013 in which the bound N is brought down to 700. Independently Tao announced on his blog on exactly the same day that he used the same method to obtain a similar reduction. Using the new method the old Polymath8 was renamed as Polymath8a and a new Polymath8b project was started. This resulted in April 2014 in bringing the bound down to 246. The bound can even be 6, but that requires to assume that the Elliott–Halberstam conjecture (1968) holds, which is a claim about the distribution of primes in arithmetic progression.</p>
<p>But Neale in this booklet brings more than just the account of this thrilling quest to close the gap. She also succeeds in explaining parts of the proofs and she also tells about similar related problems from number theory. For example the Goldbach conjecture: "every even number greater than 2 is the sum of the squares of two primes", or its weak version: "every odd number greater than 5 is the sum of three primes", are two famous examples. The generation of Pythagorean triples is another well known example. But there are other, maybe less known ones like Szemerédi's theorem proved in 1976, which proves as a special case a conjecture by Erdős and Turán: "the prime numbers contain arbitrary long arithmetic progressions". The Waring problem: "every integer can be written as a sum of 9 cubes, or more generally, as a sum of s kth powers, (where s depends on k), which triggered Hardy and Littlewood to count the number of ways in which this is possible. They proved the Waring conjecture by showing that there is at least one way of doing that. Neale also explains admissible sets which were used in a theorem proved by Goldston, Pintz and Yıldırım which was essential in proving and improving Zhang's bound on N. And there is some introduction to the prime number theorem and the Riemann hypothesis.</p>
<p>Neale cleverly interlaces these diversions with the progress on the twin prime problem, which has the effect that some tension is built up and new developments pop up as a surprise. Some of the notions and terminology that popped up in the other problems turn out to be related or at least to be useful in the twin prime problem.</p>
<p>Neale realizes that she is writing for a general audience and carefully explains all her concepts. However, I can imagine that some of the mathematics, like for example the formulas for the asymptotics in the Hardy-Littlewoord theorem involving a triple sum, fractional powers, complex numbers, and gamma functions will be hard to swallow for some of her readers. On the other hand, many of her "proofs" rely on visual inspection of coloured tables, and she has witty ways of explaining some concepts. For example admissible sets are presented as punched cards, a strip with a sequence of holes at integer distances, and the idea is that when this is shifted along the line of equispaced integers, then at least one (or more) primes should be visible in the punched holes. Modulo arithmetic she explains using a hexagonal pencil with the numbers 1-6 printed on its sides at the top, then 7-12 next to it etc. If you put the 6 sides of the pencil next to each other, you get a table of numbers modulo 6, and the primes in this table show certain patterns. Some of the graphics are less functional, yet very nice. On page 6 where prime and composite numbers are explained, a prime number p is represented with p dots lying on a circle, while composite numbers are represented by groups of dots arranged in doublets, triangles, squares, etc. which gives a visually pleasing effect. Other graphics are referring to a pond with frogs, grasshoppers, ducks, reed and waterlily leaves. These may be less instructive, but they are still a nice interruption.</p>
<p>Vicky Neale has accomplished a great job, not only in bringing the mathematics and the mathematicians to a broad audience. We meet some of the great mathematicians of our time like Gowers and Tao, both winners of the Fields Medal. We are informed how mathematical progress works, how new ideas are born. This can be through novel communication channels such as the Polymath, but it can still be a loner who works on a completely different approach who comes up with a breakthrough. Sometimes we can gain from results slumbering in mathematical history, but often it relies on coincidences when someone connects two seemingly unrelated results. And when the time for an idea is ripe, then it happens that two mathematicians independently from each other come up with the same result simultaneously.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>The book brings an accessible account about the progress that was made in the period 2013-2014 in attempts to solve the twin prime conjecture. It also sketches the way in which mathematicians think and collaborate, for example through a new communication channel such as the Polymath projects which are online blogs promoted by Timothy Gowers and Terence Tao, two prominent mathematicians, both winners of a Fields Medal.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/vicky-neale" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Vicky Neale</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/oxford-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">oxford university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-1987-8828-7 (hbk) </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£ 19.99 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">176</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Number Theory</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://global.oup.com/academic/product/closing-the-gap-9780198788287" title="Link to web page">https://global.oup.com/academic/product/closing-the-gap-9780198788287</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/11-number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11 Number theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/11-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/11a41" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11A41</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/11b25" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11B25</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/11n13" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11N13</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/11p05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11P05</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/11p32" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11P32</a></li></ul></span>Tue, 20 Feb 2018 18:22:30 +0000Adhemar Bultheel48284 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/closing-gap#commentsThe Riemann Hypothesis. A Million Dollar Problem
https://euro-math-soc.eu/review/riemann-hypothesis-million-dollar-problem
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a translation of the Dutch book <em>De Riemann-hypothese: Een miljoenenprobleem</em> published by Epsilon Uitgaven in 2011. It grew out of an intensive online course given by the authors in the period 2006-2010 for talented secondary school students. The course lasted four weeks and students got support for solving the many exercises via the internet.</p>
<p>
The idea of the course is fully maintained in this text. It has four parts, corresponding to the four weeks of the course. Its material is accessible for interested secondary school students and, why not, for starting university students as well. It is however not a leisurely reading text. Serious working and solving the many exercises that are sprinkled throughout the text is required. No more online support for the book but solutions are provided in an appendix at the end. For readers who are in for a challenge, there are somewhat more demanding exercises at the end of each part. Three other short appendices refer to external sources. One briefly gives a sketch of why large prime numbers are important. Basically it just mentioning that they are used in the RSA cryptosystem. Another one refers to freely available software packages. Besides the commercial algebra packages like Maple and Mathematica, the free software proposed can be found on the web page of Wolfram Alpha. The necessary commands to solve some of the exercises are given too. For more intensive computations, one is referred to the Sage website. A third appendix lists four books and a number of websites for further reading and experimenting.</p>
<p>
The contents of the four parts does not bring big surprises. The first one of course has to start with prime numbers and introduces the prime counting function $\pi(x)$, counting the number of primes less than $x$. Some experiments to approximate this staircase soon leads to the idea that logarithms must be involved. The prime number theorem $\pi(x)\sim x/\log(x)$ soon pops up, but a sketch of the proof has to wait till the end of the book. Another choice is Chebyshev's function $\psi(x)=\sum_{p\le x}\lfloor\log_p x\rfloor \log p$ where the sum is over the primes $p$. Because in this one, the primes are weighted depending on the number of their powers less than $x$, this $\psi(x)$ is almost a straight line. This way of weighting the primes when counting them is of course is an essential element in the analysis of Riemann. So a sneak preview of the hypothesis is the end of the first part.</p>
<p>
In the second part, the key player is the Riemann zeta function $\zeta(x)=\sum_{k\in\mathbb{N}} k^{-x}$. In order to introduce this properly, a discussion is needed to define infinite sums and functions defined by power series. In order to evaluate $\zeta(2)=\pi^2/6$, an infinite product for the sinc function is derived. The cliffhanger for this part is Euler's product formula that links the zeta function to the primes.</p>
<p>
A sketch of the proof of this Euler formula $\zeta(x)=\prod_{p~\mathrm{prime}} 1/(1-p^{-x})$ is the start of part three. The zeta function is however taking over again since it needs to be extended to the whole complex plane. This requires a crash course on complex numbers and complex functions. An elementary form of analytic continuation allows to define $\zeta(z)$ for all complex $z\ne1$. The end of this part is again a forward reference to the next one announcing the trivial and nontrivial zeros of $\zeta(z)$. With these defined, it finally becomes possible to fully understand the meaning of the formulation of the Riemann hypothesis: all the nontrivial zeros of the zeta function are on the critical line $\mathrm{Re}(z)=1/2$.</p>
<p>
In part four all the efforts come to a conclusion. The $\psi(x)$ function can be expressed as $x-\ln(2\pi)$ plus some correction. And using Euler's formula, the correction can be expressed as a sum over the zeros of the function $\zeta(z)$. The trivial zeros $-2k$ are easily obtained via Riemann's functional equation and the part in $\psi(x)$ corresponding to these trivial zeros can be summed up to give $−\frac{1}{2}\ln(1−x^{−2})$. So the remaining sum is related to the nontrivial zeros, which is the core issue of the Riemann hypothesis. The book culminates in a proof of the prime number theorem along the lines of the proofs by Hadamard and de la Vallée Poussin by showing that all the nontrivial zeros are strictly inside in the critical strip $0< |z|< 1$. </p>
<p>
It is clear that the text is quite a challenge for secondary school students, but with some elementary introductions to topics that do not belong to their standard curriculum, they are brought a long way on the road to understand the Riemann hypothesis. Although infinite cosine series do appear in the text, the text stops on the verge of where Fourier analysis needs to take over. At least, Fourier analysis is not formally introduced. That is where Mazur and Stein in their version of <a href="/review/prime-numbers-and-riemann-hypothesis" target="_blank">Prime Numbers and the Riemann Hypothesis</a> push the limit a bit further. It is a marvelous idea to bring young students this far on the scale of mathematics. What I am a bit missing is the importance of the Riemann hypothesis. Explaining the RSA encryption with some details would of course requiring another booklet of this type, but just mentioning it briefly is not really bringing the insight or making the importance of proving the hypothesis very concrete. But of course one has to draw the line somewhere, and there are other popular books around where one can read more about RSA and other wonderful things about prime numbers. This booklet is a wonderful guide when teachers around the globe want to stimulate the interest in mathematics or explain what pure mathematicians in the 21st century are working on. Perhaps they might think of starting up a similar course as the authors of this book did. The latter claim that several of the students that attended their course afterwards decided to start a mathematics education at the university.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is the English version of a booklet that the authors originally published in Dutch after they had given an intensive online course on this subject. They organized it four years in a row for interested secondary school students. Their course (and also this text) requires hard working since many exercises are provided that should be solved (the book gives solutions at the end) to properly assimilate the material. It brings the readers to a level of understand what the Riemann hypothesis is. They even prove the prime number theorem. All the mathematics that does not belong to the standard curriculum of that age (infinite series, complex analysis,...) is provided as far as needed for their purpose.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/roland-van-der-veen" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Roland van der Veen</a></li><li class="vocabulary-links field-item odd"><a href="/author/jan-van-de-craats" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jan van de Craats</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/maa-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">MAA Press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9780883856505 (pbk) </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£32.00 (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">155</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Number Theory</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.maa.org/press/ebooks/the-riemann-hypothesis" title="Link to web page">http://www.maa.org/press/ebooks/the-riemann-hypothesis</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/11-number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11 Number theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/11m26" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11M26</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/11a41" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11A41</a></li></ul></span>Sat, 04 Mar 2017 11:23:25 +0000Adhemar Bultheel47499 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/riemann-hypothesis-million-dollar-problem#commentsPrime Numbers, Friends Who Give Problems: A Trialogue with Papa Paulo
https://euro-math-soc.eu/review/prime-numbers-friends-who-give-problems-trialogue-papa-paulo
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Paulo Ribenboim is a number theorist, born in Brazil in 1928, who is living in Canada since 1962 where he was professor at Queen's University. In the tradition of the Socratic dialogues, he wrote this trialogue on prime numbers in which he is Papa Paulo and his opponents are Eric and Paulo. These two are interested in prime numbers. In fact, it is Eric who starts asking elementary questions about primes to which Papa Paulo answers. Soon Eric is joined by his friend Paulo (the other Paulo), and Papa Paulo is renamed to be P.P. (for obvious reasons he is not really happy with that alias). I believe Ribenboim wants to picture Eric and Paulo as young adults, and this is how I first imagined them, also since P.P. is addressing them as such, but then they seem to be very knowledgeable about many things (for example they point out to P.P. that Édouard Lucas was French or Eric who is said "to have travelled a lot"), things you would not expect from teenagers asking elementary questions about prime numbers. Whatever they are, it is just a minor glitch in the story, which does not affect the mathematics.</p>
<p>
The discussion thus starts at a very elementary level but after a while it gradually turns into a course on prime numbers with formulas, computations, theorems and proofs. There are some intermissions in italic like <em>`Eric paused for a while, then continued'</em>. Here is another one: P.P. tells that Fermat after his death meets Saint Peter who has to decide on whether he should be sent to Heaven, Hell, or Purgatory. Fermat is confronted with his little lie about the short proof that he had for his last theorem but that the margin was too small to contain it. In that chapter P.P. is discussing the primality of Fermat numbers and states at the end that it is not known whether there are infinitely many Fermat numbers that are prime or that are composite. Then that chapter ends with the funny remark: `<em>The effect of this strong statement of ignorance caused this reaction on Paulo and Eric: Poor Fermat, he may stay in purgatory forever.</em>'</p>
<p>
The latter illustrates that the conversation that has mostly a serious mathematical aspect, also has instances with funny components. Besides these few italic parentheses and some notes at the end of the chapters in which some biographical notes are added about a person that was mentioned (Euclid, Euler, Mersenne, Legendre, Fermat, and many many more), the whole book is just reproducing the conversation among the three protagonists. There is another bit of an unrealistic aspect to this trialogue when it comes to all the computations and formulas or formulations of theorems with their proofs. The latter formulation include the titles `Theorem' and `Proof' in bold and end with an q.e.d. message. This is something you only find in a printed mathematics book, not in a conversation, unless the discussion is taking place while the characters are writing down what they are saying as it is printed. This is indeed how we should read it because at some point P.P. <em>says</em>: `You are sharp-eyed, but what I <em>wrote</em> is correct.' (my emphasis). So he <em>wrote</em> it, not <em>said</em> it. Although P.P. is in fact to be identified with the author (Ribenboim is the meta-P.P.), he basically only reproduces the conversation and does not tell us much of the meta-story about the who, how, what, and where of the actors outside what is in the conversation. So there is only a very thin sketch of their personality, and only few circumstantial remarks in the conversation go besides the mathematical discussion. Ribenboim is just following the literary genre of the Platonic dialogues seasoned with contemporaneity and humour.</p>
<p>
Although the reading is light, the book is not easy for a truly unskilled reader since, as the book advances, the mathematics get more and more involved. In the beginning it is about the Euclidean algorithm, gcd, lcm, modular arithmetic, the Wilson theorem, Fermat numbers, and Mersenne primes, up to primality testing and public key encoding. But when it comes to the prime number theorem, it requires real numbers, the log and exp functions and the logarithmic integral and for the formulation of the Riemann hypothesis, complex numbers and complex functions, series, analytic continuation and much more advanced mathematics need to be introduced. Nevertheless, the `technical stuff' is left out as much as possible. At some point, one of the intermissions read: <em>Papa Paulo was visibly happy with the presentation of the important theorem of Dirichlet on primes in arithmetic progressions. He was particularly elated to have been able to hide all the technical innovation needed to prove the theorem in its general form...'</em></p>
<p>
Towards the end of the book many more curious facts and conjectures about prime numbers are formulated (twin primes and the likes, conjectures by Goldbach, Sophie Germain, Bunyakovskii, Schinzel and Sierpinski, and many others). There are even conjectures by Papa Paulo and by Eric.</p>
<p>
Paulo Ribenboim has written some dozen books almost all published by Springer. This one is published by World Scientific, so I do not know how much is fiction and how much is truth, but the last chapter is about publishing the notes of the trialogue. P.P.'s usual (fictional) publisher Marcel Spank at Gold Springs Publishing Company New York does not like the original title <em>Prime Experiments Explained to Boys and Girls</em> and proposes <em>The Story of Two Boys in Love with Prime Numbers</em> (from this it should be understood that Eric and Paulo are indeed boys and not adults). Eventually Spank turns down the manuscript. This at the time of P.P. writing this chapter it is still uncertain whether the notes will be published or not. It is also in this chapter that P.P. gives his unconventional idea about why there are so few women in mathematics: He, being a man, gets his best ideas while shaving, and women don't shave, hence....</p>
<p>
As a conclusion, I liked the book and at some stages it is absolutely funny. The reader should however be prepared to swallow all the mathematics, the theorems, the proofs, the formulas and all the computations. As long as only integers are involved, in principle anybody motivated enough can understand what is going on. When it becomes more involved around the formulation of the prime number theorem, it may become a bit more difficult to hang on, but then it becomes interesting again when all these mysterious properties about prime numbers are conjectured. I can imagine that it will get smart young people interested in starting a career involving number theory.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Two boys, Eric and Paulo, start asking questions about prime numbers to Papa Paulo. The conversation between those three grows into an introduction to number theory, in particular to the properties of primes and all the interesting questions and conjectures that can be formulated about them. The whole story is told in the form of a trialogue, but it involves theorems and proofs as well. It is intended for a broad audience, and yet it gives an introduction to what the Riemann conjecture is all about.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/paulo-ribenboim" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Paulo Ribenboim</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2016</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-981-4725-80-4 (pbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£36.00 (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">336</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Number Theory</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.worldscientific.com/worldscibooks/10.1142/9836" title="Link to web page">http://www.worldscientific.com/worldscibooks/10.1142/9836</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/11-number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11 Number theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/11a41" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11A41</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/97f60" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97F60</a></li></ul></span>Fri, 02 Dec 2016 11:45:40 +0000Adhemar Bultheel47308 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/prime-numbers-friends-who-give-problems-trialogue-papa-paulo#commentsThe “Golden” Non-Euclidean Geometry
https://euro-math-soc.eu/review/%E2%80%9Cgolden%E2%80%9D-non-euclidean-geometry
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Fibonacci numbers and their relation to the golden ratio are among the few mathematical items that gained some publicity among non-mathematicians. The golden ratio ($\phi=1.68033...$) is well known since antiquity and it played an important role in Euclid's <em>Elements</em> and in the work of many other mathematicians. It also shows up in phylotaxis and spirals that appear in nature. And it relates to harmony, another term that has been studied in a mathematical sense since the Greek. The golden ratio has therefore gained some mythical and even mystical status, the latter often has to be understood in a (pejorative) numerological sense.</p>
<p>
Fibonacci numbers (denoted $F_n$) is a term coined by Édouard Lucas in the 19th century, who also introduced the sequence of Lucas numbers (denoted $L_n$). Both sequences are solutions of the difference equation $x_{n}=x_{n-1}+x_{n-2}$. The initial conditions for the Fibonacci sequence are $(x_1,x_2)=(1,1)$, while for the Lucas numbers it is $(x_1,x_2)=(1,3)$. The limit of ${x_n}/{x_{n-1}}$ equals $\phi=({1+\sqrt{5}})/{2}$ in both cases. These numbers are defined for all integer indices by $x_{-n}=(-1)^nx_n$.</p>
<p>
In a first chapter, the authors give a brief historical survey, summarize some properties of the Fibonacci and Lucas numbers and they introduce hyperbolic functions: the symmetric Fibonacci hyperbolic sine $sFs(x)=({\phi^x-\phi^{-x}})/{\sqrt{5}}$ and cosine $cFs(x)=({\phi^x+\phi^{-x}})/{\sqrt{5}}$. Similarly for the Lucas versions $sLs(x)$ and $cLs(x)$ but these do not have the denominator $\sqrt{5}$. Their graphs look very much like the graphs of the standard hyperbolic functions.</p>
<p>
The second chapter is about harmony. The old Greek <em>Music of the spheres</em> was picked up by Pacioli and Kepler. But soon the text comes down to one of Stakhov's pet horses, namely that harmony is a forgotten pillar in mathematics. Counting and classical measure theory are the two other pillars that have resulted in conventional mathematics. However by rejecting Cantor's axiom (a 1-to-1 correspondence between the reals and the points on a line) and a consistent application of the golden ratio and its generalizations, a different measure theory, number system, and geometry can be developed. This is what he calls harmonic mathematics. He considers a delayed version of the above difference equation which leads to the introduction of a new representation of number systems and his $p$-Proportion Codes. However this is soon replaced by another generalized Fibonacci sequence, defined for any real $\lambda>0$ by $F_\lambda(n+1)=\lambda F_\lambda(n)+F_\lambda(n-1)$, with $F_\lambda(0)=0, F_\lambda(1)=1$ and the limiting ratio $\phi_\lambda=(\lambda+\sqrt{4+\lambda^2})/2$ which is a root of the characteristic equation $x^2-\lambda x-1=0$. The above Fibonacci hyperbolic functions can be generalized by replacing $\phi$ by $\phi_\lambda$ and the $\sqrt{5}$ by $\sqrt{4+\lambda^2}$. They are denoted as $sF_\lambda$ and $cF_\lambda$. Note that (up to a factor 2) the classical hyperbolic functions are obtained as a special case of the $\lambda$-Lucas numbers by choosing $\lambda=e-1/e$. For $\lambda=1,2,3,4$ we get the golden, silver, bronze, and copper relations, referred to as the metallic relations.</p>
<p>
The third chapter is about Hilbert's fourth problem, in which it is asked to design new forms of non-Euclidean geometry. The formulation was however rather vague and different proposals were made but it remained unclear whether the problem was (completely) solved or not. So the authors have their own interpretation and solve their form of the fourth problem using the hyperbolic functions introduced above. Lobachevsky's hyperbolic geometry is based on classical hyperbolic functions. Replacing the classical ones by the hyperbolic $\lambda$-Fibonacci functions they get different hyperbolic geometries. To obtain a similar generalization for spherical geometry, yet another type of Fibonacci functions are needed. There are of the form $SF_\lambda(x)=c_\lambda\sin(x\ln\phi_\lambda)$ and $CF_\lambda(x)=c_\lambda\cos(x\ln\phi_\lambda)$ with $c_\lambda=2/\sqrt{4+\lambda^2}$. The $\ln\phi_\lambda$ factor appears here for the sake of harmony. A similar form can be obtained in the hyperbolic case giving a true hyperbolic geometry in harmony mathematics. They consider many more relations and formulas in this context and claim that the Clay Mathematics Institute made a mistake by not putting Hilbert's fourth problem on their list of millennium problems. So the authors claim to have actually solved a self declared millennium problem.</p>
<p>
The next chapter 4 is about the qualitative theory of dynamical systems based on harmony mathematics. Hence the `golden' and also the other metallic proportions show up again. It is a simple observation that a metallic ratio $\phi_\lambda$ (which is an irrational number) can be approximated from above and below by ratios of successive $\lambda$-Fibonacci numbers. This simple fact is exploited in a complicated framework of foliations of a 2D manifold. First foliations of such a manifold are introduced, which is then specialized to the 2D torus $T^2$. These foliations are characterized by a Poincaré rotation number $\omega$. In the particular case that it happens to equal a metallic proportion, then it can be approximated by ratios of Fibonacci numbers and hence the irrational foliation is approximated by rational ones. Since integral curves for flows of a dynamical system are foliations, this may also be applied in a context of dynamical systems. This chapter is much more mathematical with long mathematical proofs which do not seem to be easily accessible for a general public.</p>
<p>
A last chapter is about the fine structure constant in physics. Like the mathematical millennium problems, there is a list of physical millennium problems. The first of these problems is asking whether all dimensionless parameters of the physical universe are calculable. Here the fine structure constant $\alpha$ is declared to be fundamental and hence is the constant to be discussed. The approach taken here is by looking at the Lorentz transform in special relativity theory. It is a transformation of the space-time vector whose matrix can be written as a direct sum of the identity and a hyperbolic rotation over an angle $\theta\in(-\infty,\infty)$. In view of the preceding items it is again a natural thing to replace the classical hyperbolic sine and cosine functions of the rotation angle by the hyperbolic Fibonacci sine and cosine ($\lambda=1$) of an appropriate angle $\psi$ and so obtain a Fibonacci special relativity theory. Here however $\psi\in(-\infty,0)\cup(2,\infty)$ because singularities appear at 0 and 2. Moreover, the speed of light in vacuum has to be made variable. It decreases with the age of the universe. It will be $c^*$ (the classical value) for $\psi\to-\infty$ and it is $c^*/\phi^2$ for $\psi\to-\infty$. The physical meaning is that the Big Bang corresponds to $\psi=0$, the interval (0,2) is the dark age before galaxies were formed (the speed of light is imaginary), and for values larger than 2 this corresponds to the light age, when the stars were formed that created light in the universe. To the left of the origin is the black hole situation with the arrow of time reversed.<br />
In 2000, N.V. Kosinov proposed a formula $\alpha=10^{-43/20}\times\pi^{1/260}\times \phi^{7/130}$. Inspired by this formula, the authors propose to let $\alpha$ depend on $\psi$ by replacing the $\phi$ in this formula by their $\psi$-depending speed of light. The result is an $\alpha(\psi)$ with $\psi=\lambda_0 T$ where $T$ is the age of the universe (in billions of years) and $\lambda_0$ a constant. This $\alpha$ is decreasing with $\psi$ in the black hole range until it becomes 0 at the Big Bang. In the same range the speed of light drops from $1/\phi$ to 0. In the dark age, the derivative is positive and goes from 0 to $\infty$ just like the modulus of the speed of light does, and in the light age it drops from infinity to a little bit below its current value of about $7.29\times 10^{−3}$. Of course as a consequence of the varying $\alpha$, also other values that depend on it will change with the age of the universe. In an appendix allusion to multiverses is made when the $\phi$ in the previous setting is replaced by $\phi_\lambda$ with $\lambda\ne 1$.</p>
<p>
The first author Alexey stakhov is a Ukrainian mathematician with a PhD in computer science, who lives in Canada since 2004. He has published many papers and books in which he has proposed many of his original, sometimes controversial, ideas. Chapter 2 clearly summarizes some of his previous work. The second author is Samuil Aranson who is a Russian mathematician, now living in the USA whose domain is differential equations, geometry and topology. It is therefore clear that he must be the main author for chapters 3 and 4, which also explains the somewhat different and more mathematical style. Scott Olson is a professor of philosophy and religion in the USA, who wrote a book on the golden section and who seems to be helping with the English editing of this book.</p>
<p>
The first two chapters are elementary with a lot of history and simple mathematical relations. Who wants to read more on Fibonacci and Lucas numbers and generalizations can read <a href="http://www.euro-math-soc.eu/review/pell-and-pell%E2%80%93lucas-numbers-applications">Pell and Pell-Lucas Numbers with Applications</a> for a good mathematical treatment and there are of course many popular books on the golden ratio. If you are interested in the golden ratio and harmony, you would certainly want to read <a href="http://www.euro-math-soc.eu/review/fibonacci-resonance-and-other-new-golden-ratio-discoveries">The Fibonacci Resonance and other new Golden Ratio discoveries</a>. However chapters 3 and 4 of this book are much more mathematical and create a complicated mathematical framework of foliations, not suitable for a general public anymore, while it only illustrates and applies the fact that the ratio of two successive Fibonacci numbers tend to the golden ratio and hence that this irrational number can be approximated by rationals. The fifth chapter is devoted to physics. The core idea is to replace a classic hyperbolic rotation by a more general one. The physical interpretation is certainly not mainstream and is probably susceptible to critique by theoretical physicists, if they do not consider it to be just numerological mysticism. However, since there is no experimental proof of what is exactly happening at this cosmological scale, it may be another explanation that is as good as many other fantasies. It is clear that the book is mainly collecting results that the authors have published as papers and that are here somewhat streamlined into a more consistent survey. Long lists of references are added after each chapter with many papers of the authors but several are only available in Russian. That this harmony mathematics and Fibonacci numbers and generalizations can solve all these problems clearly adds to the myth of the golden ratio. The typesetting in LaTeX is nicely done. I could spot a few typos but not that serious. For example page 121, a $(dv)^2$ is missing in the equation and on page 232 the Black Hole should correspond to $-\infty<\psi<0$ and not $0<\psi<2$. Also the graphics of chapter 5 are a bit rough and not always very precise. Anyway there are some original ideas to be found in this book. Whether the reader will agree with them or not will depend on who's reading it.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a book in which the authors give a summary of some of their work. They study Fibonacci and Lucas numbers and show how these give rise to a new kind of mathematics: the mathematics of harmony. Generalizations of these number sequences and their limits the golden and other metallic ratios are applied to derive a new kind of non-Euclidean geometry, to study foliations and dynamical systems and even a golden Fibonacci version of the special relativity theory in which the fine structure constant from cosmology is analyzed.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/alexey-stakhov" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Alexey Stakhov</a></li><li class="vocabulary-links field-item odd"><a href="/author/samuil-aranson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Samuil Aranson</a></li><li class="vocabulary-links field-item even"><a href="/author/scott-olsen" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Scott Olsen</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2016</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-981-4678-29-2 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£98.00 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">308</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Number Theory</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.worldscientific.com/worldscibooks/10.1142/9603" title="Link to web page">http://www.worldscientific.com/worldscibooks/10.1142/9603</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/11-number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11 Number theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/11b39" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11B39</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/53a35" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">53A35</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/37d40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">37D40</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/83a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83A05</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/83f05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83f05</a></li></ul></span>Fri, 23 Sep 2016 08:30:58 +0000Adhemar Bultheel47182 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/%E2%80%9Cgolden%E2%80%9D-non-euclidean-geometry#commentsPi: The Next Generation
https://euro-math-soc.eu/review/pi-next-generation
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
In <em>Pi. A source book</em> the editors L. Berggren, J. Borwein and P. Borwein, assembled a number of reprints that sketch the history of pi, its mathematical importance and the broad interest that it has received through the centuries from the Rhind papyrus till modern times. The last edition (3rd edition, 2004, to which I will refer as SB3) added several papers that related to the computation of the digits of pi by digital computers. Rather than extending this with more recent developments (SB3 was already some 800 pages), it was decided to collect this computational aspect in a new volume. This "<em>The next generation</em>" volume got the rightful subtitle "<em>a source book on the recent history of pi and its computation</em>". Because it extends the papers on digital computation that were added in SB3, the trailing papers of SB3 are reprised here. The papers are ordered chronologically, so of the first 14 papers in this book, 12 were already at the end of SB3.</p>
<p>
It starts with the agm (algebraic-geometric mean) iteration attributed to Salamin and Brent who both published their papers in 1976. It generates two sequences of numbers by iteratively extending the sequences respectively with the algebraic and the geometric mean of the previous numbers. Given appropriate initial conditions, both sequences converge to a common limit related to pi. This method is widely used since these publications of 1976, but the agm idea was actually used already by Gauss and others although not in connection with computing pi. The Borwein brothers discuss a quartically convergent method based on it (1984) and Bailey and Kanada used it to compute millions of decimals of pi (1988). The number of digits computed today has exceeded these computer experiments by many orders of magnitudes and several papers in this book survey the history, and the diversity of formulas and methods and the successive records reached.</p>
<p>
There are, besides the classical methods to compute pi, also several computational methods to generate the expansion of pi. For example, a completely different spigot algorithm computes the decimals of pi one by one but using only integer arithmetic (originally from 1995 and extended in 2006). In a more classical vein is the BBP algorithm (named after the authors Bailey, Borwein and Plouffe) which allows to compute a set of binary (or hexadecimal) digits of pi without the need to compute all the previous ones (1997). This is of course a great help when computing trillions of digits. Of course there are a a number of papers devoted to Ramanujan's notebooks with formulas to compute pi.</p>
<p>
There are also some papers on the proof of irrationality of pi, and of related numbers such as its roots, ζ(2), ζ(3); (i.e. Apéry's constant), Catalan's constant etc. The investigation of the properties of the digits of pi, in particular the normality of pi (still unproved) is discussed and computationally tested. The tests can be nicely visualised using random walks and color coding. Normality means that every possible sequence of <em>m</em> successive digits is equally probable for any basis and for any <em>m</em>.</p>
<p>
The papers are reprinted in their original format, thus with different fonts, lay-out, etc. It happens that the end of a previous chapter or article is still on the first page of the reprint or the start of the next one is on the last page. Even some totally unrelated announcement that appeared at the end of the original journal paper, it is reprinted here unaltered. Just as one would in a pre-digital age collect photocopies of the papers. Nevertheless, the book has an overall name and subject index, which is not obvious in this case. Since the papers come from many different journals (and even some chapters of a book) not all of these papers may be readily available or even known to an interested researcher, or in this case, it may even be a lay person who is interested. Many of the papers have authors that are the main players in the field: David Bailey, Bruce Berndt, and Jonathan and Peter Borwein. As this book was being printed one of its editors, Jonathan (Jon) Borwein, passed away on 2 August 2016. So it was probably too late to add a dedication or a note in this book. This collection he helped to compile and containing several papers that he coauthored, can be considered one of his last gift to the scientific community. </p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a collection of reprints of 25 papers discussing pi. Mostly about the computations of its digits and checking the normality. They are ordered chronologically from 1976 to 2015. This is an alternative for yet a fourth edition of <em>Pi. A Source Book</em> by Berggren and the Borwein brothers, the third edition of which appeared in 2004. The computational papers of that 3rd edition are reprised as the initial papers of this volume.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/david-h-bailey" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David H. Bailey</a></li><li class="vocabulary-links field-item odd"><a href="/author/jonathan-m-borwein" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jonathan M. Borwein</a></li><li class="vocabulary-links field-item even"><a href="/author/eds-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(eds.)</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/springer-verlag-0" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">springer verlag</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2016</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9783319323756 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">74.19 € (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">521</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Number Theory</a></li><li class="vocabulary-links field-item even"><a href="/imu/numerical-analysis-and-scientific-computing" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Numerical Analysis and Scientific Computing</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/gp/book/9783319323756" title="Link to web page">http://www.springer.com/gp/book/9783319323756</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/11-number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11 Number theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/11-04" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11-04</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/11y16" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11Y16</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/68q25" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68q25</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/11k16" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11K16</a></li></ul></span>Tue, 23 Aug 2016 07:39:47 +0000Adhemar Bultheel47119 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/pi-next-generation#commentsSumming It Up: From One Plus One to Modern Number Theory
https://euro-math-soc.eu/review/summing-it-one-plus-one-modern-number-theory
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This book is the third in a trilogy by these authors who want to introduce number theory to the `mathematically literate reader'. The first two books were <a href="http://www.euro-math-soc.eu/review/fearless-symmetry-exposing-hidden-patterns-numbers"><em>Fearless Symmetry: Exposing the Hidden Patterns of Numbers</em></a> (2006) and <a href="http://www.euro-math-soc.eu/review/elliptic-tales-curves-counting-and-number-theory"><em>Elliptic Curves: Curves, Counting and Number Theory</em></a> (2012). The first dealt with Diophantine equations and Fermat's Last Theorem (FLT), and the second with elliptic curves and the Birch-Swinnerton-Dyer (BSD) conjecture. Both of these somehow ran ashore at the end when bumping into modular forms which were not fully explained. This third book <em>Summing It Up: From One Plus One to Modern Number Theory</em> has as its main objective to introduce the reader to these modular forms and their applications. Like the previous books it has three parts. Parts one (finite sums) and two (infinite sums) are further motivations and at the same time also preparations for the main part three about modular forms</p>
<p>
Number theory in its simplest form is easy to understand even for the man in the street and innocent looking questions or conjectures can be formulated, and yet answering or proving them is extremely difficult and may take even centuries and the brightest mathematicians to answer them. The good thing about this is that, in trying to answer these questions, whole new mathematical areas are explored that lead us far away from the simple question about integers. Modular forms is such an instrument from complex analysis with major application in number theory, but that also appear for example in string theory and algebraic topology.</p>
<p>
It is not a simple task to explain to the lay person that a modular form of weight <em>k</em> is a holomorphic function of the upper half plane of the form $f(z)=a_0+a_1q+a_2q^2+\cdots$, $q=\exp(2\pi i z)$, bounded as $\mathrm{Im}(z)\to\infty$ and that satisfies $f(\gamma(z))=(cz+d)^kf(z)$ for any $\gamma$ in the modular group, i.e., $$\gamma\in\mathrm{SL}_2(\mathbb{Z})=\left\{\left[\begin{array}{cc}a &b\\c&d\end{array}\right]:a,b,c,d\in\mathbb{Z}, ad-bc=1\right\}\quad\text{defining}\quad\gamma(z)=\frac{az+b}{cz+d}.$$ And yet, the authors succeed in gradually leading the reader to this level and connect this to elliptic curves and number theory. They have to guide the reader through complex analysis, hyperbolic geometry, introduce the concept of groups, fundamental domains, and analysis of the modular group and their traditional generators and congruence subgroups. To make sure the reader is following up to that point, it requires strong motivation, and of course the reader should not be completely unfamiliar with mathematics. In fact the mathematical skills required are building up as one is reading on. With only a first calculus course in your backpack, it can be done, but you will need some patience and perseverance to reach the end.</p>
<p>
Part 1, requires only high school algebra and some geometry. This part on finite sums introduces modulo calculus and the type I (equal 1 modulo 4) and type III (equal 3 modulo 4) numbers, which are important to know when a positive integer is the sum of two squares. The latter problem is then gradually generalized to: find out how many ways there are to write a positive integer $n$ as a sum of $k$th powers. Another topic treats summation techniques and formulas to add a finite number of $k$th powers, including binomials and Bernoulli polynomials.</p>
<p>
For the second part the reader needs to know differentiation, infinite series, and Taylor expansion as it can be found in a first year calculus course. Besides the infinite sums (and products), the reader is also is prepared for part three with complex numbers and some complex functions, especially the symbol $q=e^{2\pi i z}$ is important in for modular forms. Furthermore it also introduces analytic continuation, the zeta function, generating functions and Dirichlet series.</p>
<p>
The third part is the most important and takes about half of the book. Since the previous part introduced the necessary material, hence provided it sank in well enough, no extra knowledge is required. As mentioned above it leads the reader to the concept of a modular form. But it goes beyond just the definition. What if subgroups of $\mathrm{SL}_2(\mathbb{Z})$ are considered, and how do these relate to the weight <em>k</em> of the modular form? And how are modular forms related to generating functions for a sequence of numbers? For example the sequence $\{p(n)\}_{n=1,2,...}$ where $p(n)$ represents the number of partitions of $n$, i.e., the number of different ways in which $n$ can be written as a sum of positive integers. The generating function for this sequence can actually be written as an infinite product, which, like in the case of the zeta function, makes it interesting. Here the link with previous topics can be made: consider a sequence $\{a(n)\}$ as considered in part 2. I.e., $a(n)$ expresses the number of different solutions to a problem depending on a positive integer $n$. If these are the coefficients in a generating function or Dirichlet series, then modular forms can be used to investigate their properties. But not only the applications in number theory, also applications in other areas (some of them are mentioned in the trailing chapter) make modular forms an interesting research area <em>an sich</em>.</p>
<p>
There is some mathematics indeed: there are definitions, theorems, proofs, but the most complicated proofs are left out. There are even some occasional exercises for the reader to work out, but the text is not really structured in a strict textbook kind of way. It is more a written-out oral presentation with a lot of explanation and intermittent questions like: "such and such is true. Why? Well remember that..." or "What happens now if..." or "Is that good enough? Yes!" etc. There are also reasonings that simulate the reader's thoughts in the style of "Let's try ... But that does not work if... So we need to..." until all pitfalls are removed. Only after this `trial and error' intro the eventual correct definition is formulated. Nevertheless, it is not the hand-waving kind of popularization. There is true rigorous mathematics and the reader should be prepared to swallow it. Anyway the authors did a remarkable job in making some aspects of modern number theory very accessible to readers with only a minimal knowledge of mathematics, say a student who had a first calculus course. However, also mathematicians who do not have number theory as their main focus will enjoy this book although they will probably skip some of the sections dealing with the introduction of elementary topics such as summation formulas, groups, complex numbers, or vector spaces. They will learn how complex analysis and group theory (joining forces in modular forms) are nowadays working tools to tackle some questions in modern number theory.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a third book that these authors wrote providing an introduction to some aspects in number theory for readers with only a limited knowledge of mathematics. The first two parts on finite and infinite sums respectively serve as a further motivation and also as an introduction to the main objective of the book: an introduction to modular forms.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/avner-ash" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Avner Ash</a></li><li class="vocabulary-links field-item odd"><a href="/author/robert-gross" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Robert Gross</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2016</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9780691170190 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 27.95 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">248</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Number Theory</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://press.princeton.edu/titles/10692.html" title="Link to web page">http://press.princeton.edu/titles/10692.html</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/11-number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11 Number theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/11f03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11F03</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/20h05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">20H05</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/11f06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11F06</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/11f11" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11F11</a></li></ul></span>Tue, 05 Jul 2016 14:04:40 +0000Adhemar Bultheel47039 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/summing-it-one-plus-one-modern-number-theory#commentsPrime Numbers and the Riemann Hypothesis
https://euro-math-soc.eu/review/prime-numbers-and-riemann-hypothesis
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The Riemann hypothesis is currently, now that Fermat's Last Theorem has been proved, the unsolved problem in mathematics that has been researched most, both theoretically and experimentally. Many known 'theorems' start with 'If the Riemann hypothesis holds then...' and often it also holds the other way around. Thus there are many equivalent formulations. Contrary to FLT, its formulation is not that easy to understand for the non-mathematician. The authors have chosen to give several formulations of the hypothesis, starting from the most direct elementary form: how many prime numbers are there less than a certain number, and gradually formulating equivalent forms that are more and more mathematical: from the staircaise function $\pi(x)$, counting prime numbers up to $x$, until the well known formulation about the location of the zeros of the zeta function.</p>
<p>
The authors have written the book in four parts. The first one takes about half of the booklet and goes through the historical development from elementary prime number concepts to Riemann's use of Fourier analysis to understand the spectrum of the prime number distribution. This is intended for any interested reader. Mathematics are not or only present in disguised form. All the difficult technicalities and the frightening sharp edges are nicely hidden. The reader is however treated with dignity, i.e., (s)he is not considered as a complete idiot. All what is needed is some interest in learning more about the problem at hand. For example it is explained how the logarithms enters the scene, what the logarithmic integral is, and how it was used by Gauss to approximate the prime number distribution, and how in Fourier analysis one represents an arbitrary function as a summation of cosines.</p>
<p>
Part II is described by the authors as a preparation to extensions of Fourier analysis as needed in the next parts for readers who had at least one calculus course. This is indeed needed in the next part where the link is made between the location of the prime numbers and the Riemann spectrum. The step is not trivial. Approximating a smooth function by a sum of cosines is one thing, but approximating a staircase function stepping at all the integer muptiples of prime numbers requires distributions, which is itself already a difficult concept. But the reader is convinced that the idea works, not so much by the theory but by the graphics of experiments that show the spikes appearing at the appropriate places where primes or their powers should appear. To appreciate part IV the reader is assumed to have some knowledge about complex functions because it comes to the final description of the hypothesis as the statement about the location of the nontrivial zeros of the Riemann zeta function on the $x=1/2$ axis in the complex plane. Here one needs to introduce the concept of analytic continuation for the summation of infinite power series with complex exponents, and to link this with infinite products involving primes. Especially when one has to link zeros of the zeta function to the spectrum introduced before, it may become a bit fuzzy for the reader that is not properly prepared.</p>
<p>
Although there are already many references in the text, there is a kind of appendix with endnotes which give further references, often links to an internet site or to a pdf where the full paper that is referred to can be downloaded. These notes are also used to give extra technical explanation. At several places these are really essential, certainly in the later parts, if you do want to get to the mathematics.</p>
<p>
The chapters are very short, sometimes just one paragraph, so that the reader is brought to the next level teaspoon by teaspoon. And yet the reader is introduced to random walks, Cesàro summation, and to Fourier analysis, but also to distributions, and how they can be used in Riemann's Fourier approach. There are also many graphics clearly showing the approximations for the staircase of primes and how these look at different scales. These are essential in the concept of the book. They strongly contribute to the understanding of what is going on. The zeta function comes surprisingly late into the picture, or maybe not so surprising since this is as far as the authors want to bring the reader.</p>
<p>
Besides the graphs, there are many other illustrations of the main mathematicians involved, historical as well as contemporary. The booklet is published on glossy paper. It is not recommended to buy the paperback edition because the pages easily get detached from the cover, a very unfortunate property, since very soon you will end up with a set of loose pages and a separate cover instead of a nice book. Since the book is thin enough, a hard cover does not seem a good option either. However, the authors did a wonderful job. Given its compactness and the richness in content, this is a marvelous booklet. It does exactly what the authors intended to: introduce the reader to the problem. You will absolutely not find here a springboard to the mathematics needed to solve the problem. Thus you will not learn how to tackle the problem, in fact nobody currently knows how to solve it, but you will learn about the standard mainstream approach so far. So not how to proceed in the future, but a short history and an idea about the what and why of the Riemann Hypothesis is expertly explained. Whether or not you have some mathematical background, you can pick the level that suits you.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This wonderfully illustrated booklet explains for the lay person in very slow pace what the Riemann hypothesis actually is, why it is important, and what kind of partial results are known so far. It starts with the most elementary concepts of what prime numbers are and builds up to the full formulation of the hypothesis. It will depend on the mathematical background of the reader how far she will be able the keep up with the mathematics.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/barry-mazur" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Barry Mazur</a></li><li class="vocabulary-links field-item odd"><a href="/author/william-stein" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">William Stein</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/cambridge-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">cambridge university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2016</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9781107499430 (pbk), 9781107101920 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£17.99 (pbk), £39.99 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">154</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Number Theory</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9781107101920" title="Link to web page">http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9781107101920</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/11-number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11 Number theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/11m26" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11M26</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/11a41" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11A41</a></li></ul></span>Tue, 21 Jun 2016 10:48:52 +0000Adhemar Bultheel47012 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/prime-numbers-and-riemann-hypothesis#commentsNumbers: Histories, Mysteries, Theories
https://euro-math-soc.eu/review/numbers-histories-mysteries-theories
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This booklet is the English translation of the German original from 2013: <em>Zahlen, Geschichte, Gesetze, Geheimnisse</em> The book starts and ends with the question: What is a number? Beutelspacher gives answers by sketching the historical evolution of the concept of numbers from counting one-two-three-many up to complex numbers.</p>
<p>
The author is a popularizer of mathematics, well known in Germany through his columns, and his many books. He is also the founder of the <em>Mathematikum</em>, a math museum in Giessen. The present book, his most recent, tells the evolution of the concept of numbers, and their representation through the centuries. The idea is that the text should be accessible for anyone with no or just a minimal knowledge of mathematics. This subject has been covered by many other authors before. For example Havil's <a href="/review/irrationals-story-numbers-you-cant-count"><em>The irrationals</em></a> or Stewart's <a href="/review/professor-stewarts-incredible-numbers"><em>Professor Stewart's incredible numbers</em></a> or Ifrah's monumental <em>The Universal History of Numbers</em>, and there are of course many more. So what is new here? Well, it is short, and yet nothing essential has been left out and it is truly explaining all that is needed for the layman to understand what is going on. However, it certainly is not a flat executive summary because it still has details and anecdotes to keep the attention of the reader.</p>
<p>
The content is organized in five chapters. That it should start with the integers is obvious. In fact the first chapter starts from counting in a primitive society. Gradually the concept of a natural number emerges and the early mathematicians investigated number patterns like even and odd numbers, square and triangular numbers, magic squares, and Pythagorean triples, but also prime numbers. A brief excursion is made to Fermat's last theorem and the ultimate proof by Wiles. Some cryptography and the basic idea of RSA coding are explained. This shows how important natural numbers still are in our modern society.</p>
<p>
The second chapter deals with the representation of numbers. The origin is of course tallying, and different notations and number systems. There were the Egyptian and Roman systems, which were not so useful for computing. The Babylonians had a place value sexagesimal system, which was much more useful. We inherited our 60 minutes in an hour and 60 seconds in a minute. For example their number 234 could for example denote 2 hours, 3 minutes and 4 seconds. It is however via the Arabic mathematicians that the Indian decimal system as we know it, including the zero, was introduced in Europe. The chapter also gives a good explanation of how the abacus was used for computing. Also some divisibility rules are explained. Division by 2, 5 and 10 are trivial of course, but still, the check digit at the end of our EAN barcodes is based on the remainder modulo 10 of the weighted sum of the digits in the code. Finally, there is obviously the binary system. It was described by Shannon in 1948 as the basis for communication, although Leibniz envisioned already a binary computer, but he did not elaborate on it.</p>
<p>
The story of the rationals and irrationals is told in the next chapter. The step to be made from the geometric concept of proportion to the ratio of two integers and then to the rational number that this ratio represents is not so obvious. The Egyptians had an ingenious system of unit fractions to compute with, but the true concept comes again from the Indians and it can be connected with decimal representation of numbers (containing a finite number of digits). But rationals had their limitations and gradually, starting with the incommensurability problem of the Greek, the irrationals conquer their way into the minds of mathematicians. The golden ratio which appears in the pentagram, was already scrutinized by the Greek. The marvelous proof of the irrationality of the square root of 2 is included. It is also pointed out that there are algebraic irrationals and transcendental irrationals.</p>
<p>
Chapter 4 prolongateso this idea and continues the dissection of the transcendentals. That requires the introduction of limits of number sequences. The rational numbers are now extended with the limits of these sequences. This gives for example the result that 0.999... represents a limit that is actually the same as 1, a fact not so easily accepted by a general reader. We are further instructed about the approximations to pi by the Greek, the introduction of Euler's number e and we are introduced to Cantor's theory of the infinite and how his diagonal technique could prove that there are infinitely many transcendental numbers</p>
<p>
The imaginary and complex numbers became necessary when one wanted to solve polynomial equations. We learn how al-Khwārizmī solved quadratic equations with geometric constructions, while Cardano had a formula which made him believe that a solution to such an equation can also be a negative number. In a geometric context, numbers are lengths, and then a negative solution is not acceptable. The algebra made possible what geometry could not deliver. For the cubic equation, we get the story of the Tartaglia-Fior duel and how Cardano pilfered Tartaglia's formula so that he could publish it and that is how today it gets Cardano's name attached to it. The quintic equation bears the dramatic story of Abel and Galois who both died at a young age. Abel from pulmonary tuberculosis and Galois from the consequences of a physical duel over a love affair. This stopped the race to find algebraic formulas for the solution a polynomial equation of higher degree. This doesn't mean that there are no solutions to the equation. Already quadratic equations required complex numbers, but they were not recognized. A quadratic with complex roots was considered to have none. It was Cardano who first used complex numbers implicitly, but without recognizing them in his computations for the cubic. After complex numbers were accepted, the fundamental theorem of algebra stating that every polynomial equation of degree <em>n</em> has <em>n</em> real or complex roots was soon formulated. It was however Gauss who finally proved it almost two centuries after Roth had given a first hint.</p>
<p>
The conciseness of the text and the objective of readability for laymen, necessitates some loose formulations that are strictly speaking not a hundred percent correct when isolated from the context. For example `Every equation can be solved!' (p.85) means actually `Every polynomial equation has a complex solution', or '[The binary system] is the representation of numbers used by modern computers' (p.37) which is only partially true since they work mostly with the hexadecimal number system with a lot of bells and whistles attached. And there are more such examples. But these are of course nitpicking and clearly, when seen in context, the lay reader will certainly have no problem with such formulations. To make every sentence unambiguous would only harm readability. It is only when they are used as a section title (like the one on page 85) that makes a mathematician frown.</p>
<p>
The text is pleasant to read and is well illustrated. The reader is gently guided through number wonderland, avoiding any abstraction or complexity that could deter the innocent reader. And yet, in all its simplicity such a reader is still challenged to follow the author in some of his scratching the surface of algebraic equations or some other true mathematical issues that are a bit more technical. Let me end by noting that the translators (A. Bruder, A. Easterday, J.J. Watkins) did an excellent job. In fact they have added an appendix with additional notes that contain sometimes more details (e.g. they give a proof of the nine test), often they refer to other (mostly recent) books or even websites where more information can be found. A useful addition because it is quite acceptable that this guided tour gets some readers hooked who want to read more about this fascinating world of numbers.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This booklet is a translation of the German original of 2013 including some additional notes. It introduces the reader with a minimum of mathematical knowledge to the story of how the concept of a number evolved through the historical development of mathematics. From the counting numbers to the integers, rationals, irrationals, transcendentals, and complex numbers. </p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/albrecht-beutelspacher" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Albrecht Beutelspacher</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/dover-publications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Dover Publications</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2016</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0486803487 (pbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">US$12.95 (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">112</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Number Theory</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://store.doverpublications.com/0486803481.html" title="Link to web page">http://store.doverpublications.com/0486803481.html</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/11-number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11 Number theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/11-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11-03</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-01</a></li></ul></span>Sun, 13 Mar 2016 13:41:44 +0000Adhemar Bultheel46796 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/numbers-histories-mysteries-theories#commentsConvolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms
https://euro-math-soc.eu/review/convolution-and-equidistribution-sato-tate-theorems-finite-field-mellin-transforms
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book is a specialized work on sums over finite fields. This algebraic topic was begun by Gauss when he considered sums over prime fields of the product of non-trivial additive and multiplicative characters of the field. From that starting point, the situations where similar sums appear are ubiquitous in Number Theory. For example, they can be found in studies devoted to the four square problem, or bounds for the number of solutions of some polynomials on finite fields.</p>
<p>A remarkable situation behind these objects is the distribution of the value of these sums when the size of the field goes to infinity. It turns out that they are uniformly distributed with respect to a convenient measure, a result which required a substantial investigation and that is connected with other algebraic contexts. The author is a world experto on this topic, with remarkable contributions in the literature. In fact the book under review is not the first contribution of Prof. Katz in Annals of Mathematics Studies devoted to equidistribution of sums or related problems. The interested reader can find a list of them in the webpage of Princeton University Press. In this book, the author tackles the questions in broad generality, giving a general statement on equidistribution in the modern language of sheaves.</p>
<p>The organization of the book is as follows. The first chapter motivates the problem and gives the statement of the main results of the book. Chapters 2 to 7 provided the constructions needed for these main results, which are enriched with complementary results in chapters 8 to 12. The rest of the chapters present a interesting variety of examples illustrating the theory whereas the final part of it addresses the situation in the case of the ring of integer numbers.<br />
The final result is a complete book, with beautiful and important results on equidistribution properties of sums built from subtle and complicated techniques borrowed from Algebra and Geometry. This book is mainly aimed at experts on the field as well as advanced readers interested in algebraic number theory.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Marco Castrillon Lopez</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book is devoted to the distribution of sums of characters on finite fields when the size of the fields goes to infinity. The author, a world expert on the topic, presents the problem in broad generality using techniques from algebraic geometry.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/nocholas-m-katz" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Nocholas M. Katz</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2012</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-691-15330-8</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">203</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/algebra" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Algebra</a></li><li class="vocabulary-links field-item odd"><a href="/imu/algebraic-and-complex-geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Algebraic and Complex Geometry</a></li><li class="vocabulary-links field-item even"><a href="/imu/number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Number Theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/11-number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11 Number theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/11t24" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11T24</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/11g25" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11G25</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/14f04" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">14F04</a></li></ul></span>Tue, 22 Dec 2015 18:45:27 +0000Marco Castrillon Lopez46627 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/convolution-and-equidistribution-sato-tate-theorems-finite-field-mellin-transforms#commentsThe Fibonacci Resonance and other new Golden Ratio discoveries
https://euro-math-soc.eu/review/fibonacci-resonance-and-other-new-golden-ratio-discoveries
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The book deals with the Golden Ratio $\phi$, Fibonacci numbers $F_n$ and friends. And there are a lot of friends. The Fibonacci numbers satisfy $F_n=F_{n-1}+F_{n-2}$ with $F_0=0$ and $F_1=1$ and $\phi=(1+\sqrt{5})/2$ which is the limit of $F_{n+1}/F_n$. For this book, the Binet relation $F_n=(\alpha^n-\beta^n)/\sqrt{5}$ with $\alpha=\phi$ and $\beta=1/\phi=(1−\sqrt{5})/2$ is even more important. Throughout history in human artwork, the Golden Ratio 1:$\phi$ has repeatedly appeared, but also 1:$\phi^s$ with $s$ an integer (positive or negative) or one half. Another important player is the logarithmic spiral (the Golden Spiral) with polar equation $r=\phi^{b\theta}$ with $b=2/\pi$. This means that with every quarter circle, the value of the radius is an integer power of $\phi$ anchored at $\phi_0=1$ for $\theta=0$. Lucas numbers $L_n$ satisfy the same recurrence $L_n=L_{n-1}+L_{n-2}$ but they start with $L_0=2$, and $L_1=1$. Their Binet formula is $L_n=\alpha^n+\beta^n$. It is supposed that $F_{-n}=(-1)^{n+1}F_n$ and $L_{-n}=(-1)^nL_n$ for all integer $n$.</p>
<p>
All this is pretty well known, and it can of course be found in this book, but there is much more. It attempts to give a roundup of all that is known about Fibonacci numbers and friends, and adds to this a new insight from the author: the Fibonacci resonance. As the author states in his preface, this volume is bundling three books in one. The first one (here represented by Part I) is conform to what most popular science books mention about these issues. It is an historical survey of how $\phi$ and the Fibonacci numbers appeared in science, in nature, and in artwork in the past. The `third book' (appearing in the form of Part V) is a continuation of the first one, but these elements are less easily found in the literature, at least not in this accessible form, since it discusses several applications of $\phi$ and friends in more recent developments of science. The middle piece (Parts II, III, IV) is more mathematical and develops some original ideas of the author about what he calls the Fibonacci resonance which is based on the elements that were introduced in my first paragraph of this review. But let me start with the historical background.</p>
<p>
All the usual suspects appear in the historical survey. Of course the pyramids from Egypt, but also, and these may be less familiar to an average public, the megalithic Sun and the Moon gates in Bolivia, which are extensively discussed. Also the meter of classical Sanskrit involves mathematical patterns. Obviously on the list are also the mathematics of the ancient Greek, re-introduced in the West by the Arabs, and the scientific evolution since the 15th century with the naming of the Fibonacci numbers and the introduction of the symbol $\phi$. The spirals are introduced and their classical appearance in nature (nautilus, pineapple, pine cone, sunflowers, Roman broccoli, etc.). Also in music such patterns can be detected. Bartók, Debussy, and Xenakis are taken as examples. Paris became 'the capital of $\phi$' in the course of the 19th and the early 20th century when artists picked up the Golden Section credo that was propagated by scientists and theoreticians who strongly influenced the Parisian art scene. Among them Charles Henry, friend of the mathematician Édouard Lucas who studied the Fibonacci sequence, Joséphin Péladan who promoted $\phi$ on mystical grounds, Maurice Princet, the mathematician of cubism etc. Extensive discussions are devoted to paintings of Seurat, Toulouse-Lautrec, and Mondrian, the purist `par excellence'. Architecture is represented by Le Corbusier and his Modulor, a system of proportions based on an anthropomorphic scaling.</p>
<p>
The `third book' is called '$\phi$ science' and is also 'traditional' in the sense that it lies in the line of expectations. Here we meet the recent applications of the Fibonacci sequence. There we obviously find phylotaxis, not only in plants, but also in DNA, superconductors, and sunlight harvesting. A link not mentioned so far is sphere packing and tiling. Here we meet 3D crystal structures, Penrose tilings, and quasicrystals (extensively discussed), Islamic patterns, superlattices and composites (metamaterials) with the protagonists (Victor Veselago, John Pendry, Dan Shechtman) and applications (cloaking, plasmonics,...). Of course all these applications are practically important, but some topics such as Fibonacci word and Penrose tiling are interesting structures that invite to be studied at an abstract theoretical level.</p>
<p>
This leaves us with the most original, most surprising, and most mathematical part of this book. The first idea is to divide the goniometric circle into 32 equal parts. This results in an Ori32 geometry referring to 32 possible orientations. Menhinick got his inspiration from the fact that many applications depend on angles and orientations, rather than the classical point-line-plane approach to geometry. With these 32 parts, the smallest part of the disk is thus a wedge with angle $\pi/16$. This is used as a kind of unit and is called a MIK. It is denoted as $\fbox{1}$ and one can consider multiples of this unit. A quarter disk corresponds for example to $\fbox{8}$. The disk can be divided into wedges progressing in Fibonacci-like manner. Putting together $\fbox{1}$, $\fbox{2}$, $\fbox{3}$, $\fbox{5}$, $\fbox{8}$, and $\fbox{13}$, one get the full disk because $(1+2+3+5+8+13)\pi/16=2\pi$. Because of the periodicity, calculus with these MIK is modulo 32. If the radius of the circle is 1, then the arc length of $\fbox{5}=5\pi/16\approx1$. Similarly the arc of $\fbox{8}=\pi/2\approx\phi$ and for $\fbox{3}$ we get $3\pi/16\approx1/\phi$ etc. All these approximations are too large, so we may forget $\fbox{1}$ (which contributes $\phi^{-3}$) and get $2\pi\approx\phi^{-2}+\phi^{-1}+1+\phi+\phi^2=2\phi+3=4+\sqrt{5}$. Because the arc lengths are only approximately correct, Menhinick points to the analogy of the Pythagorean comma in music theory. His search for the necessary correction resulted in a generalization of the Binet formula: $F_n=F_s\alpha^{n−s}+F_{n-s}\beta^s$ for all integers $n$ and $s$. To visualize this, a golden spiral with equation $r=(F_s/\phi^s)\phi^{b\theta}$ is constructed for each $s$. These are in fact simple transforms of a (standard) golden spiral. The first term $F_s\alpha^{n−s}$ defines points on the spiral at its intersections with the coordinate axes. The second term is a quantized deviation to get $F_n$ from the previous spiral points namely $F_{n-s}$ times a quantum $\beta^s$. Similarly for the Lucas numbers, one has $L_n=\sqrt{5}F_s\alpha^{n-s}+L_{n-s}\beta^s$. The $\sqrt{5}$ rotates the axes over about 150 degrees and the intersections of these rotated axes with the spirals gives again approximations $\sqrt{5}F_s\alpha^{n-s}$ for $L_n$ with a quantized deviation $L_{n-s}\beta^s$. Note that each spiral has its own characteristic quantum $\phi^{-s}$. Then Menhinick considers the finest quantum to be half a wavelength. Since quanta for different $s$ always appear in integer multiples, this can be considered as standing waves of different frequencies that 'vibrate' in resonance: the Fibonacci resonance. It is also investigated whether there is fractal behavior but that issue seems not to be cleared out completely. This is followed by an extra part in which these ideas are applied to generalized Lucas sequences $U_n=PU_{n−1}+QU_{n-2}$, starting with initializations 0 and 1. The Pell and Pell-Lucas numbers are a special case for $(P,Q)=(2,−1)$. The analog of $\phi$ is here the Silver Ratio $\delta=1+\sqrt{2}$. They were recently (2014) studied in the book <a href="/review/pell-and-pell–lucas-numbers-applications" target="_blank"><em>Pell and Pell-Lucas Numbers with Applications</em></a> by T. Koshy, which is not in the (otherwise quite extensive) list of references of this volume.</p>
<p>
All the material that I discussed so far takes about 400 of the approximately 600 pages. The remaining one third of the book consists of appendices with technical and mathematical details, glossaries of terms and symbols used, a collection of formulas, and most of all an overwhelmingly extensive list of references (1004 items!). Also the index is well stuffed and useful for an encyclopedic work such as this book.</p>
<p>
There is no doubt that what is described as the first and the third book are useful additions to what is already available in the literature. There are certainly original contributions also there. About the second book, introducing Fibonacci resonance, I am not so sure where all this is leading to, and what to think of all this spiraling number magic. There are obviously interesting, and as far as my knowledge is concerned, new relations derived in this part. What I mean to say is that I can certainly appreciate the formulas underlying the concept, but the resonance interpretation hints to numerological significance that I believe unnecessary. A remark such as the fact that the right-hand side of $\sum_{n=−2}^7\phi^n=\frac{11}{2}(7+3\sqrt{5})$ (formula (12.3)) combines the first 5 primes: 2,3,5,7,11, is of course true, but it is in my opinion pure coincidence and has no further meaning. And there are other examples, including the resonance interpretation, which is amusing and imaginative, but otherwise with little mathematical significance.</p>
<p>
The illustrations are plentiful and helpful, except perhaps the 3D model of the different $s$-spirals which is for me only more confusing than what is already in the previous chapter. All the facts and persons of the book are extremely well researched and referenced. Also the pointers forward and backward are detailed and make it so much easier for the reader, and the typesetting in LaTeX is practically flawless (some italic instead of roman <em>log</em>'s and <em>arctan</em>'s here and there are minor exceptions).</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is an encyclopedic work about the history, theory, and applications of the Golden Ratio and the Fibonacci numbers (and their companions, the Lucas numbers). A remarkable addition to this is the middle part of the book in which Menhinick develops a theory of Fibonacci resonance that is based on recursion formulas for Fibonacci and Lucas numbers expressing them as deviations form certain points on logarithmic spirals.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/clive-n-menhinick" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Clive N. Menhinick</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/onperson-international-ltd" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">OnPerson International Ltd.</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2015</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0993216602 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">US$ 89.00</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">638</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-science-and-technology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics in Science and Technology</a></li><li class="vocabulary-links field-item even"><a href="/imu/number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Number Theory</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.amazon.com/Fibonacci-Resonance-other-Golden-discoveries/dp/0993216609" title="Link to web page">http://www.amazon.com/Fibonacci-Resonance-other-Golden-discoveries/dp/0993216609</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/11-number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11 Number theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/11b39" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11B39</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/11-00" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11-00</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a20</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/00a65" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a65</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a66" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a66</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/00a67" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a67</a></li></ul></span>Tue, 08 Dec 2015 09:38:20 +0000Adhemar Bultheel46596 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/fibonacci-resonance-and-other-new-golden-ratio-discoveries#comments