Experimental Number Theory
This booklet is intended for graduate students in pure mathematics with a number-theory orientation, and for those who incline to learn mathematics through experiments. Its main goal is neither to give an introduction to number theory, nor to learn a programming language, but to show how the computer can be used for numerical experimentations in number theory in the spirit of J. W. S. Cassels’ motto “Number theory is an experimental science”. Sample programs are written in the PARI-GP language and are available to download from the author's website.
The book is written in an easily readable style, giving the reader basic information on the computed issues, as well as a sufficient amount of facts for further orientation in the surrounding theory. The examples draw from the theory of the quadratic reciprocity law, special sequences (like Bernoulli numbers, trinomial ones, etc.), sums of squares or binary quadratic forms, zeta functions of varieties over finite fields, elementary class field theory, modular forms, p-adic analysis, to mention just a few. The book can be warmly recommended not only for those with a preference for number theory but even for the general readership in pure mathematics or supervisors looking for creative examples for further motivation of scientific computations.