Greek Number Theory

Abstract

In the first chapter we saw that number theory has been important in the history of mathematics for at least as long as geometry, and from a founda-tional point of view it may be more important. Despite this, number theory has never submitted to a systematic treatment like that undergone by elementary geometry in Euclid’s Elements. At all stages in its development, number theory has had glaring gaps because of the intractability of elementary problems. Most of the really old unsolved problems in mathematics, in fact, are simple questions about the natural numbers 1, 2, 3,…. The nonexistence of a general method for solving Diophantine equations (Section 1.3) and the problem of identifying the primes of the form 2^2h + 1 (Section 2.3) has been noted. Other unsolved number theory problems will be mentioned in the sections that follow.