Fermat's problem, also called Fermat's last theorem, has attracted the
attention of mathematicians for more than three centuries. Many clever
methods have been devised to attack the problem, and many beautiful
theories have been created with the aim of proving the theorem. Yet, despite
all the attempts, the question remains unanswered.
The topic is presented in the form of lectures, where I survey the main
lines of work on the problem. In the first two lectures, there is a very brief
description of the early history, as well as a selection of a few of the more
representative recent results. In the lectures which follow, I examine in suc-
succession the main theories connected with the problem. The last two lectures
are about analogues to Fermat's theorem.
Some of these lectures were actually given, in a shorter version, at the
Institut Henri Poincare, in Paris, as well as at Queen's University, in 1977.
I endeavoured to produce a text, readable by mathematicians in general,
and not only by specialists in number theory. However, due to a limitation
in size, I am aware that certain points will appear sketchy.
Another book on Fermat's theorem, now in preparation, will contain a
considerable amount of the technical developments omitted here. It will
serve those who wish to learn these matters in depth and, I hope, it will
clarify and complement the present volume.