After Pythagoras proved that A^2 + B^2 = C^2 for right triangles with hypotenuse C, many mathematicians asked if there were non-zero integers such that A^n +B^n = C^n for n>2. By the third century, it was a common conjecture that no such solution was possible for any n>2. In 1637, Pierre de Fermat wrote in his copy of Diophantus's Arithmetica ``I have discovered a truly remarkable proof which this margin is too small to contain." This ``result" became known as Fermat's Last Theorem, yet no proof of his was ever found, only a correct proof for n=4. In 1993, corrected in 1995, Andrew Wiles proved something much stronger from which the truth of Fermat's Last Theorem came as a corollary. There still is no direct proof of Fermat's Last Theorem. 

This talk will focus on the history of attempts to prove Fermat's Last Theorem, some of the fields of mathematics which started in this pursuit, a common proof technique with roots in Fermat's proofs for low n, and conclude with a modern example of an attempt to prove a special case using only what Fermat knew, where it went deceptively wrong, and how to find such mistakes yourself. The talk will be accessible to undergraduates, graduate students, and faculty. All are welcome.