"Eular's Theory" by Ajit Mishra's Online Classroom

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The interest of Leonhard Euler (1707–1783) in number theory was first spurred in 1729, when a friend of his, the amateur Goldbach pointed him towards some of Fermat's work on the subject. This has been called the "rebirth" of modern number theory, after Fermat's relative lack of success in getting his contemporaries' attention for the subject. Euler's work on number theory includes the following:
  • Proofs for Fermat's statements. This includes Fermat's little theorem (generalised by Euler to non-prime moduli); the fact that \scriptstyle p = x^2 + y^2 if and only if \scriptstyle p\equiv 1\; mod\; 4; initial work towards a proof that every integer is the sum of four squares (the first complete proof is by Joseph-Louis Lagrange (1770), soon improved by Euler himself); the lack of non-zero integer solutions to \scriptstyle x^4 + y^4 = z^2 (implying the case n=4 of Fermat's last theorem, the case n=3 of which Euler also proved by a related method).
  • Pell's equation, first misnamed by Euler. He wrote on the link between continued fractions and Pell's equation.
  • First steps towards analytic number theory. In his work of sums of four squares, partitions, pentagonal numbers, and the distribution of prime numbers, Euler pioneered the use of what can be seen as analysis (in particular, infinite series) in number theory. Since he lived before the development of complex analysis, most of his work is restricted to the formal manipulation of power series. He did, however, do some very notable (though not fully rigorous) early work on what would later be called the Riemann zeta function.
  • Quadratic forms. Following Fermat's lead, Euler did further research on the question of which primes can be expressed in the form \scriptstyle x^2 + N y^2, some of it prefiguring quadratic reciprocity.
  • Diophantine equations. Euler worked on some Diophantine equations of genus 0 and 1. In particular, he studied Diophantus's work; he tried to systematise it, but the time was not yet ripe for such an endeavour – algebraic geometry was still in its infancy. He did notice there was a connection between Diophantine problems and elliptic integrals, whose study he had himself initiated.

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