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Number Theory: An approach through history from Hammurapi to Legendre

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When someone like Weil sets out to write a history of number theory it is destined to be the standard reference for decades to come. But this is not only an authoritative reference everyone loves to cite--it is also delightfully readable. It is not a substitute for a textbook (although Weil hints at this possibility is the preface), but even for readers with only a modest background in number theory this book will be a source of insight and joy.

Chapter 1 "Protohistory" treats briefly some of the scattered pre-Fermat attempts, which helped form the Diophantine tradition of what would constitute the staple problems of number theory--Pythagorean triples, sums of squares, Pell's equation, such things. These seeds blossomed in the hands of Fermat (chapter 2), with whom we start to see the formation of a coherent theory of numbers with some basic tools: infinite descent, modulo arguments, a precursor of elliptic curve arithmetic, etc. Fermat rarely wrote things down properly, and Euler (chapter 3) had to work hard to prove his theorems and conjectures, in the process adding some ideas of his own (the group theoretic core of modulo arithmetic and Z/pZ, auxiliary functions such as the phi function, etc.). Euler's further investigations along these lines also left many valuable ideas for future mathematicians such as the crystallisation of the importance of quadratic forms (taken up by Lagrange, chaper 4; later perfected by Gauss) and the statement of the law of quadratic reciprocity (taken up by Legendre, chapter 4; later proved in full by Gauss). Also highly decisive for the future development of number theory was Euler's bringing in of analytic ideas into number theory, in particular elliptic integrals (whose deep importance was later revealed by Jacobi) and the zeta function and L-series (whose deep importance was later revealed by Dirichlet and Riemann).

This item is of very great interest to me. It covers thoroughly, in brilliant exegesis, the whole history and mathematics of number theory from Hammurapi to Legendre at a level suitable for the general reader with an interest in number theory. The author was one of the pioneers in this field.