Fermat’s Last Theorem??

Who was Fermat? Born in France Became city councilor in Toulouse Then he became a judge Had a reputation for being distracted Died in 1665

Fermat’s Distractions Discovered math at University of Bordeaux Influenced greatly by the greeks Developed methods to find maxima, minima and tangents Laid the foundation for Calculus Invented a form of coordinate geometry Never published anything- wrote many letters to friends and other math geeks

The important stuff Fermat also tooled in number theory Often wrote notes in the margins of books he was reading 5^2= 3^2+4^2 Thus a square is written as a sum of two squares And Fermat says….

Fermat’s Last Theorem “In contrast it is impossible to divide a cube into two cubes, or a fourth power into fourth powers, or in general any power beyond the square into powers of the same degree. I have discovered a marvelous proof of this, but this margin is too narrow to contain it.”

So what your really trying to say is Z^n = x^n + y^n has not solution if n>2. Fermat’s son Samuel published these notes in In the 1700s Leonhard Euler went through most of Fermat’s number theory and proved it. He even found a mistake! Euler proved Fermat’s last theorem for n=3.

The issue Without a proof, Fermat’s last theorem is not a theorem. No one ever found Fermat’s “marvelous proof” Two parts of the proof No solutions when x,y,z are not divisble by n No solutions when one of x,y,z is divisble by n

Sophie and Kummer Sophie Germain proves a generality If n is prime and 2n+1 is prime then Fermat’s last theorem has no solutions in the first case. Mathematicians continue to nibble at pieces of the proof for awhile. Kummer proves n where all primes are “regular” A regular prime is a prime that does not divide the numerator of a Bernulli number

A long time ago Not so much history anymore, Fermat’s last theorem was proven in 1993 by Andrew Wiles A few months later he announced that his proof had a hole in it Then in September 1994 the manuscripts of the fixed proof circulated 350 years later Fermat finally had a theorem

The proof Extremely complicated final proof but The big idea is to prove that n=4 has no solution. Thus any even n that is not equal to 2 is divisible by 4. Then an odd number is either prime or divisible by a prime or many primes.

Summary Fermat’s last theorem was the greatest unsolved problem in mathematics for 300 years It is important not for what was learned with the proof, but what was learned in the attempts to prove it Much of modern number theory is derived from attempts to prove the theorem Historians agree it is highly unlikely Fermat had proven his own theorem

Timeline Fermat’s Last Theorem is published 1700s- Euler proves most of Fermat’s number theory Germain’s proof is published by Legendre Dirlect and Legendre prove n= Lame proved n= Lame gets greedy and tries to use complex numbers to prove FLT Ernst Kummer disproves Lame and proves n for all regular primes FLT is proved by Andrew Wiles

The reference centre.ics.mq.edu.au/alfpapers/a142e.pdf# search=%22Applications%20of%20Ferma t's%20last%20theorem%22http://www- centre.ics.mq.edu.au/alfpapers/a142e.pdf# search=%22Applications%20of%20Ferma t's%20last%20theorem%22 t=Regular