Fermat's little theorem, ICTMT5, Klagenfurt University, Austria. August 2001.

Fermat's little theorem

For Mark Daly - former colleague and friend -
who helped me (and still does) when I was really stuck

John Cosgrave, Mathematics Department,
St. Patrick's College, Drumcondra,
Dublin 9, IRELAND.
email: [email protected]
web site: http://www.spd.dcu.ie/johnbcos

A thing of beauty is a joy for ever.
Its loveliness increases; it will never
Pass into nothingness.
[John Keats (1795-1821), Endymion (1818)]

Fermat's 'little' theorem is one of the jewels of Number Theory, and to mark the anniversary of Fermat's birth ( August 2001), I offer this talk. My talk is not intended as an introduction to Number Theory, nor indeed even as an introduction to Maple, although in both cases it could serve as such. Indeed in the time available (some 30 minutes) it will be possible for me to cover only a very small selection of the topics listed below. Anyone who is interested may access the this Maple .mws file at my web site, and also a html text conversion (which may be read by anyone who has a web browser, and does not require having Maple); they are in the Public and Other Lectures section of the Maple section of my site.

Because of the widely reported work of Andrew Wiles, many non-mathematicians have heard of Fermat's Last Theorem - that has no solutions in non-zero integers x , y and z , for , ... - and it is probable that more significant mathematical ideas have been developed in attempting to prove that result than any other single mathematical question.

But, is the Last theorem one of any consequence? Has anyone (yet!) found an application for it? Such-and-such is true because of Fermat's Last Theorem?

On the other hand his little theorem - which was relatively easy to prove (but how many could create a proof ab initio ?) - has a vast range of mathematical consequences, and one major practical application. The purpose of my talk is to identify in a single document a number of those consequences, but I am aware that I have not covered all of them; for example I have not touched upon the topic of Carmichael numbers (even though they are there in the background of Guiga's conjecture in Section 7).