Fermat, Pierre de (1601–1665)
Pierre de Fermat was a French lawyer, magistrate, and gentleman scholar, often called the Prince of Amateurs, who is best known for the conjecture, now proved, known as Fermat's last theorem. Although employed as a senior government official, Fermat somehow managed to find time to do an astonishing amount math, for which he sought little acclaim or acknowledgement. In fact, he published only one important manuscript in his entire lifetime and even then used fake initials. When Roberval offered to edit and publish some of his works, Fermat replied "whatever of my works is judged worthy of publication, I do not want my name to appear there." Most of his results are known through letters to friends, notes in book margins, and challenges to other mathematicians to find proofs for theorems he had devised.
Fermat was one of the founders, with René Decartes, of algebraic geometry, and, with Blaise Pascal, of probability theory. His work on the maxima and minima of curves and tangents to them was seen, by Isaac Newton, as a starting point for calculus. Yet his greatest love was for number theory. In 1640, while studying perfect numbers, Fermat wrote to Mersenne that if p is a prime number, then 2p divides 2p - 2. Shortly after he expanded this into what is now called Fermat's little theorem. As usual, Fermat stated "I would send you a proof, if I did not fear its being too long." His most famous statement of this form accompanied his hasty notes on the "Last Theorem."
Fermat's little theorem
Fermat's little theorem states that If p is a prime number then for any number a, (ap - a) must be divisible by p. This theorem is useful for testing if a number is not prime, though it can't tell if a number is prime. As mentioned above, Fermat didn't himself provide a proof. Leonhard Euler was the first to publish a proof in 1736, but Gottfried Leibniz left virtually the same proof in an unpublished manuscript from sometime before 1683.