amy glen: euler's contribution to number theory

Many of Euler's contributions to number theory arose from his enthusiasm for interesting questions about the integers. One particular question that caught Euler's attention was raised by the Berlin mathematician, Naude. In 1740, Naude wrote to Euler to ask in how many ways a given positive integer can be expressed as a sum of r distinct positive integers. This problem was quickly solved by Euler and, within few months, he sent a memoir on the the subject to the St. Petersburg Academy. He had created a new area of number theory, which intrigued him for many years to follow.

First Euler introduced the idea of a partition of a positive number n into r parts as a sequence,

, of positive integers such that

, where the ni are the parts. As an example, the partitions of 4 are:

1 + 1 + 1 + 1, 1 + 1 + 2, 1 + 3, 2 + 2, 4.

He then let p(n) denote the number of partitions of n into any number of parts, where p(0) = 1. In order to study the sequence {p(n)}, he introduced the concept of a generating function

, and showed that

From this, Euler then proved that the number of partitions of n into distinct parts is the coefficient of

in the series for

, and in fact, the number of partitions of n into distinct parts equals the number of partitions of n into odd parts.

The theory of partitions was a perfect subject for Euler to exercise his great skill in formal manipulation, and he went on to prove numerous important identities. Apart from partitions, Euler also realised an even wider use for power series in number theory. He stated in a letter to Goldbach that the coefficient an in the series

is the number of ways to express n as a sum of four integer squares. Thus, if it could be proved that an > 0, for all n, then Fermat's conjecture, that every positive integer is the sum of four squares, would be true. This very representation was used in the 1800s by Jacobi (1804-1851) when he used the theory of elliptic functions to prove Fermat's claim was indeed true.