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Fermat was one of the leading mathematicians of the early 17th century although not a professional mathematician. Born at Beaumont-de-Lomagne in France, he studied law and spent his working life as a magistrate in the provincial town of Castres. Although mathematics was only a spare-time activity, Fermat was an extremely creative and original mathematician who opened up whole new fields of enquiry.

Fermat's work in algebra built on and greatly developed the then new theory of equations, which had been largely founded by François Viète. With Pascal, Fermat stands as one of the founders of the mathematical theory of probability. In his work on methods of finding tangents to curves and their maxima and minima he anticipated some of the central concepts of Isaac Newton's and Gottfried Leibniz's differential calculus.

Another area of mathematics that Fermat played a major role in founding, independently of René Descartes, was analytical geometry. This work led to violent controversies over questions of priority with Descartes. Nor were Fermat's disagreements with Descartes limited to mathematics. Descartes had produced a major treatise on optics – the Dioptrics – which Fermat greatly disliked. He particularly objected to Descartes' attempt to reach conclusions about the physical sciences by purely a priorirationalistic reasoning without due regard for empirical observation. By contrast Fermat's view of science was grounded in a thoroughly empirical and observational approach, and to demonstrate the errors of Descartes' ways he set about experimental work in optics himself. Among the important contributions that Fermat made to optics are his discovery that light travels more slowly in a denser medium, and his formulation of the principle that light always takes the quickest path.

Fermat is probably best known for his work in number theory, and he made numerous important discoveries in this field. But he also left one of the famous problems of mathematics – Fermat's last theorem. This theorem states that the algebraic analog of Pythagoras's theorem has no whole number solution for a power greater than 2, i.e., the equation

aⁿ + bⁿ = cⁿ

has no solutions for n greater than 2, if a, b, and c are all integers. In the margin of a copy of a book Arithmetica of Diophantos, an early treatise on equations, he wrote: “To resolve a cube into the sum of two cubes, a fourth power into two fourth powers, or in general any power higher than the second into two of the same kind, is impossible; of which I have found a remarkable proof. The margin is too small to contain it.”

Fermat never wrote down his “remarkable proof” and the equation was the subject of much investigation for over 350 years. In June 1993 the British-born mathematician Andrew Wilespresented a proof to a conference at Cambridge in a lecture entitled “Modular forms, elliptic curves, and Galois representations.” His proof ran to 1000 pages – rather more space than Fermat's margin – and it is generally believed that Fermat, given the mathematical techniques available at the time, must have been mistaken in believing that he had a proof of the conjecture.