Fermat

1
Fermats Last

• Theorem??

2
Who was Fermat?

• Born in France- 1601
• Became city councilor in Toulouse
• Then he became a judge
• Had a reputation for being distracted
• Died in 1665

3
Fermats 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

4
The important stuff

• Fermat also tooled in number theory
• Often wrote notes in the margins of books he was
  reading
• 52 3242
• Thus a square is written as a sum of two squares
• And Fermat says.

5
Fermats 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.

6
So what your really trying to say is

• Zn xn yn has not solution if ngt2.
• Fermats son Samuel published these notes in
  1670.
• In the 1700s Leonhard Euler went through most of
  Fermats number theory and proved it. He even
  found a mistake!
• Euler proved Fermats last theorem for n3.

7
The issue

• Without a proof, Fermats last theorem is not a
  theorem.
• No one ever found Fermats 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

8
Sophie and Kummer

• Sophie Germain proves a generality
• If n is prime and 2n1 is prime then Fermats
  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

9
A long time ago

• Not so much history anymore, Fermats 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

10
The proof

• Extremely complicated final proof but
• The big idea is to prove that n4 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.

11
Summary

• Fermats 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

12
Timeline

• 1670- Fermats Last Theorem is published
• 1700s- Euler proves most of Fermats number
  theory
• 1808- Germains proof is published by Legendre
• 1825- Dirlect and Legendre prove n5
• 1830- Lame proved n7
• 1830- Lame gets greedy and tries to use complex
  numbers to prove FLT
• 1830- Ernst Kummer disproves Lame and proves n
  for all regular primes
• 1993-4- FLT is proved by Andrew Wiles

13
The reference

• http//www-centre.ics.mq.edu.au/alfpapers/a142e.pd
  fsearch22Applications20of20Fermat's20last20
  theorem22
• http//primes.utm.edu/glossary/page.php?sortRegul
  ar