# Pell and Pell–Lucas Numbers with Applications

Fibonacci numbers form a popular sequence, even known to non-mathematicians. They show up in nature and in mathematical context, sometimes quite unexpectedly. This is partly due to the very simple recurrence $x_{n+1}=x_n+x_{n-1}$ with starting values $x_1=x_2=1$. The Lucas numbers form another independent solution with starting values $x_1=1$, $x_2=3$. Among number theorists, the Pell and Pell-Lucas numbers are well known siblings of these, solving the recurrence $x_{n+1}=2x_n+x_{n-1}$, the first with initial conditions $x_1=1$, $x_2=1$ and the second with $x_1=1, x_2=3$. The name Pell numbers was coined by Euler, who abusively attributed their origin to John Pell instead of William Broucker. Neither have Pell or Lucas anything to do with the other sequence.

Thomas Koshy, now emeritus, has written several books on discrete mathematics and number theory. His *Fibonacci and Lucas numbers with applications* (Wiley, 2001) is a twin for the present one. But he has several others for example on Catalan numbers and triangular numbers, topics that also show up in the present book. It indeed contains all you ever wanted to know about these number sequences. And probably more, since it keeps surprising the reader with yet another unexpected application or connection. Koshy has chosen, like in his previous books, for a format of a textbook with a basic approach, gradually building up knowledge, definitions, properties, theorems, as it is commonly done in a calculus or algebra course for beginning university students. In fact, the level of the mathematics required is precisely that of university novices. There are many illustrations, examples and most of all many exercises to keep the reader/student alert. To quote the author: *"Pell and Pell-Lucas numbers provide boundless opportunities to experiment, explore, and conjecture; they are a lot of fun for inquisitive amateurs and professionals alike."*

Broucker was solving the Pell Diophantine equation $x^2-dy^2=k$, with *d* a positive integer and $k=1$, a challenge passed on to him by Fermat. However that equation is much older and known to early Indian mathematicians. The equation shows up when counting bricks on a triangular pile. Hence the connection with triangular numbers such as Pascal's triangle and binomial coefficients. The Pell and Pell-Lucas numbers emerge when solving the equation for $d=2$ and $k=(-1)^n$. The three term recursion of these numbers has an obvious link with continued fractions and orthogonal (polynomial) sequences. By the form of the recurrence one might expect here that Chebyshev polynomials will enter, and indeed they do. All these different approaches converging to the common center with these sequences residing at the heart of it illustrate their importance. On the other hand, since they are at the center, it also allows for extensions and generalizations in many different directions, which inhibits cross-fertilization. Another relation, not mentioned so far is the connection with Pythagorean triples like 3,4,5 satisfying $a^2+b^2=c^2$. An infinity of such triplets appear in the sequence of triangular numbers (counting bricks in a triangular pile). All what is needed to derive and understand these results is clearly explained and defined (determinants, linear difference equations, continued fractions,...).

To continue, the reader has to step up a level and shift gear a bit. Properties of finite and infinite sums and products of (ratios) of Pell and Pell-Lucas numbers are derived and that includes their generating functions. Pell walks consists of *n* consecutive connected steps on a square grid in any of the 4 directions, without retracing. Classifying and counting them gives again the familiar numbers. When also diagonal steps are allowed, one gets Delannoy numbers. By a slight modification of the Pell recursion: $p_{n+1}(x)=2xp_n(x)+p_{n-1}(x)$, we get Pell and Pell-Lucas polynomials, for which yet again many interesting properties can be derived. Because of the link with Chebyshev polynomials, the link with trigonometry will not come as a surprise. Koshy explains this in the chapter he calls Pellonometry. More combinatorial relations are derived in tiling problems and connections with graph theory.

Most of the theorems are proved in a clear way. No solutions are provided for the exercises. This is a treasure trove of relations, formulas, connections, that circle the notion of Pell numbers. There is no comparable publication having that amount of information available on this topic. It will be of great interest to number theorists, professional as well as amateurs. Perhaps targeting the latter, is the reason why Koshy has chosen for this low level approach. It could be used as a textbook, but I doubt that there is a curriculum that will put this amount of number theory on a rather specific subject in a first or second year at the university. Of course it is perfectly possible to select only some chapters to teach from.

**Submitted by Adhemar Bultheel |

**30 / Dec / 2014