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Euler's contribution to number theory

Published online by Cambridge University Press:  01 August 2016

Peter Shiu
Peter Shiu*
Affiliation:
353 Fulwood Road, Sheffield S10 3BQ, e-mail: p.shiu@yahoo.co.uk
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Extract

Individuals who excel in mathematics have always enjoyed a well deserved high reputation. Nevertheless, a few hundred years back, as an honourable occupation with means to social advancement, such an individual would need a patron in order to sustain the creative activities over a long period. Leonhard Euler (1707-1783) had the fortune of being supported successively by Peter the Great (1672-1725), Frederich the Great (1712-1786) and the Great Empress Catherine (1729-1791), enabling him to become the leading mathematician who dominated much of the eighteenth century. In this note celebrating his tercentenary, I shall mention his work in number theory which extended over some fifty years. Although it makes up only a small part of his immense scientific output (it occupies only four volumes out of more than seventy of his complete work) it is mostly through his research in number theory that he will be remembered as a mathematician, and it is clear that arithmetic gave him the most satisfaction and also much frustration. Gazette readers will be familiar with many of his results which are very well explained in H. Davenport's famous text [1], and those who want to know more about the historic background, together with the rest of the subject matter itself, should consult A. Weil's definitive scholarly work [2], on which much of what I write is based. Some of the topics being mentioned here are also set out in Euler's own Introductio in analysin infinitorum (1748), which has now been translated into English [3].

Type
Articles
Information
The Mathematical Gazette , Volume 91 , Issue 522 , November 2007 , pp. 453 - 461
Copyright
Copyright © The Mathematical Association 2007

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References

1. Davenport, H. The Higher Arithmetic (6th edn), Cambridge University Press (1998).Google Scholar
2. Weil, André Number Theory: an approach through history; From Hammurrapi to Legendre, Birkhàuser (1983).Google Scholar
3. Euler, L. Introduction to Analysis of the Infinite, Book I; (translated by Blanton, John D.) Springer-Verlag (1988).Google Scholar
4. Heath-Brown, D. R. Fermat’s two squares theorem, Invariant (Journal of the Undergraduate Society of students of Mathematics at Oxford University) (1984) pp. 3–5.Google Scholar
5. Zagier, D. A one-sentence proof that every prime p = 1 (mod 4) is a sum of two squares, Amer. Math. Monthly, 97 (1990) p. 144.Google Scholar