Pierre de Fermat (1601-1665) is probably the greatest amateur mathematician. He thought of mathematics much like a hobby. Early on in his career, he came to possess a copy of Diophantus' The Arithmetica. In his copy, he would scribble notes here and there in the margin. These were notes about theorems and then claims of his own. Every little claim that he made in this book was later proven to be true. (with the exception of his formula for prime numbers.)
Everything was proven except for one: Fermat's Last Theorem.
cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos ejusdem nominis fas est diviere: cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non capert.
(It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.)
If n>2, then a^n+b^n=c^n has no solutions in nonzero integers a, b, and c. This one proof hung around well after fermat's death in 1665 and was finally proven a little over ten years ago, on June 23, 1993 by Andrew Wiles of Princeton and Cambridge.
The story of Andrew Wiles is somewhat interesting. That's what we learned today in math history ^_^ . Throughout the course of the last 300 years of mathematicians attempting to solve the equation, no one ever got close at all. The most interesting thing about attempting to prove Fermat's Last Theorem was the mathematics that came out of it in order to prove it, like the Taniyama-Shimura conjecture for example, which stated that elliptic curves were modular. This was the key to proving Fermat's Last Theorem. Willes went to prove Fermat through Taniyama-Shimura, then Taniyama-Shimura through Frey curves, then Frey curves through Galios representations... blah blah blah. I don't know the specifics.
All of these mathematics were created, more or less, just to prove Fermat. Which raises the question: what the hell proof did Fermat have if mathematics of today had to change and grow in order to prove it. Crazy! Most people say he couldn't, and that he made it up. Even so, look at all of that Fermat inspired in mathematics. Or, maybe he didn't make it up and this guy proved that.
Anywho, that was today's very interesting lesson in math history. Tomorrow's colloquium is hosted by Jeske, who is going to talk about Pick's theorem.
Hopefully someday I'll fulfill that childhood dream of making a theorem of my own. Schlock's theorem of Geometric Explanation? Who knows.. Maybe i could just be like Fermat and 'make up' a whole bunch of crap and let everyone else prove it. Hell, that's how Frey curves came about. Interesting, no? Schlock curves.