Imagine a cantankerous amateur mathematician on his deathbed leaving in the margins of a book an assertion of a theorem. Because Pierre de Fermat, a lawyer by formal profession, corresponded, collaborated, goaded and vexed the great mathematicians of his day—no less than Newton, Descartes, and Gauss, and proposed so many original mathematical ideas, his assertion was given credibility. And the 350-year quest for a short proof, any proof, of Fermat’s Last Theorem was on.
The assertion is a simple one—reminiscent of Pythagoras’ theorem learned by any first semester algebra student. Rather than hypotenuse squared = a squared + b squared. Fermat’s Theorem is that X^n + Y^n = Z^n where n cannot equal 0 nor can n>2. His exuberant margin notes indicated there was a simple short proof. In 1999 Andrew Wiles and a team of mathematicians proved Fermat’s theorem (and incidentally collected a lot of prize money), but Wiles’ proof was not a short proof. It was more than 100 pages long.
Some people who study the history of mathematics have insinuated that Fermat never had a short proof for his theorem. It has been postulated that he was a very competitive and argumentative man who knew he was dying, so he left the margin note to vex his competition in the mathematical world. Wiles and his team had to use mathematics that had not even been invented when Fermat was alive in order to prove Fermat’s Last Theorem.
However, Fermat was somewhat of an enigma. He remained throughout his life an “amateur” mathematician, (mostly because he did not always offer up publishable proofs of his work). Peter L. Bernstein, in his book “Against the Gods”, described Fermat as “a mathematician of rare power. He was an independent inventor of analytic geometry, he contributed to the early development of calculus, he did research on the weight of the earth, and he worked on light refraction and optics. “ Fermat also corresponded with Blaise Pascal and together they developed the theory of probability after a gambler posed to them the conundrum of why if he betted on a five being rolled on a die four times, in the long run he would win, but if two dice were rolled and he was betting on double sixes the chances were much lower. Fermat’s crowning achievement was that he is credited with developing the modern theory of numbers.
If you are feeling like you want a challenge, the search is still on for a short proof of Fermat’s Last Theorem. A mathematician of my acquaintance believes it will be solved not by a mathematician but by a creative and clever individual. There is a $1 million dollar prize offered by a Texas businessman named D. Andrew Beal to any one who can offer a short proof of the theorem. http://www.ams.org/profession/prizes-awards/ams-supported/beal-prize