Fermats Last Theorem

1
Fermats Last Theorem

• Maris Yeager
• Pre-Calculus
• Tuna

2
Who is Fermat?

• Pierre de Fermat born on August 17th, 1601 in
  Beaumont-de-Lomagne, France died on January
  12th, 1665 in Castres, France.
• Received his degree in law at the University of
  Orleans in 1631
• Math wasnt his profession, it was his hobby
• Said to be the father of Number Theory, the study
  of the properties of integers
• Also a founder of Analytical Geometry, Calculus
  and Probability
• Works were published after death by his son,
  Samuel

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Fermats focus

• Studying theorems, equations, and concepts of
  Number Theory, Geometry, and Calculus
• Example - Fermats lesser theorem If p is a
  prime number and a is any positive integer, then
  ap- a is divisible by p

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Fermats Last Theorem

• Based on the Pythagorean Theorem x2 y2 z2
• Changed the exponent from 2 to 3, giving him x3
  y3 z3
• Asked if there was a combination of integers that
  would fit this equation
• Determined that in the equation xn yn zn if
  ngt2, then the equation will never work
• He wrote the margin of the book Arithmetica this,
  I have discovered a truly remarkable proof but
  this margin is too small to contain it.
• He never wrote or published his proof of this
  theorem

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Why is FLT important?

• Efforts to prove the theorem have catalyzed the
  development of other theories
• It took 350 years to find proof
• Although FLT is a useless theorem, mathematicians
  were fascinated by the challenge to prove it
• Its more of a riddle to mathematicians

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Relation to a Diophantine equation

• Diophantine equations are polynomial equations
  with one or more variable that are solved only
  with integers
• The connection of Diophantine equations and FLT
  is that the equation in question xn yn zn is
  a Diophantine equation
• This means that only integers can be plugged in
  for the variables of FLT

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Relation to Pythagoras Theorem

• Pythagoras Theorem or the Pythagorean Theorem
  says that in a right triangle the square of the
  hypotenuse equals the sum of the squares of the
  two other sides
• Seen in variable form it is x2 y2 z2
• Fermat took the PT and changed the exponents to
  3, thus formulating his theory of the equation xn
  yn zn

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How did Fermat prove this theory?

• Simon Singh said, thats what makes FLT so
  wonderful. He said he had proof, but he never
  wrote it down.
• In his copy of the book Arithmetica, he wrote in
  the margin, I have discovered a truly remarkable
  proof but this margin is too small to contain
  it.
• He never wrote or published his proof of this
  theorem

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Attempts to prove FLT

• Sophie Germain posed as a man in the 19th century
  to study math at a men-only college. She
  eventually revealed herself when she had partial,
  but not full, proof of FLT. She proved that x, y,
  and z cannot by divisible by n if nlt100
• Leonhard Euler proved with n3.
• Johann Peter Gustav Lejeune Dirichlet proved n5
  and n14.
• Gabriel Lame proved n7 in 1839.

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Possible Proof?

• Andrew Wiles, a professor at Princeton,
  discovered that FLT was connected with the
  Taniyama-Shimura conjecture, another mathematical
  anomaly. The Taniyama-Shimura conjecture is a
  theory that suggests that elliptical equations
  and modular forms, two very different math
  concepts, are actually the same thing. If one
  could prove the TSC correct, it automatically
  meant that FLT was correct also. So Wiles started
  to work on proving TSC, in complete secrecy, for
  seven years. He finally emerged in 1993 with his
  proof. After Wiles presented his proof, a team of
  mathematicians checked his 200 page proof and
  found a flaw. Wiles went back with a former
  student, Richard Taylor, to remove the mistake.

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The Final Proof

• In May of 1995, after revising and shortening the
  original proof, Wiles and Taylor presented their
  new proof in Annals of Mathematics. This was
  checked and deemed the solution to FLT. The final
  proof was over 100 pages long.

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Proof of n5

• Lejeune Dirichlet proved in 1825 that in the
  special case of n5, either x, y, or z must be
  even and one must be divisible by 5.
• There are two cases of n5
• Case 1 When the number divisible by 5 is even
• Case 2 When the even number and the one
  divisible by 5 are distinct

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Works Cited

• 1. The New Encyclopedia Britannica Micropedia.
  Vol 4, Edition 5. Fermat, Pierre de. ã 2002
  Chicago, IL. Pg. 738-739
• 2. The New Encyclopedia Britannica Micropedia.
  Vol 4, Edition 5. Fermats last theorem. ã
  2002 Chicago, IL. Pg. 739
• 3. Comptons Encyclopedia. 8, F. Fermat, Pierre
  de. ã 2002 Success Publishing Group, LTD.
  Lombard, IL. Pg. 54
• 4. World Book Millennium. 7. Grabiner, Judith V.
  Fermat, Pierre de. ã 2000 World Book Inc.
  Chicago, IL. Pg. 73
• 5. Singh, Simon. www.simonsingh.net/Fermat_QA.htm
  l Fermat Q A.

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Works Cited (cont)

• 6. Singh, Simon. www.simonsingh.net/What_is_the_Th
  eorem.html What is the Last Theorem?
• 7. Shay, David. http//fermat.workjoke.com/
  Fermats Last Theorem. ã 2003
• 8. OConnor, JJ. http//www-gap.dcs.st-and.ac.uk/
  history/HistTopics/Fermat's_last_theorem.html
  Fermats Last Theorem. ã 1996
• 9. Sloman, Jim. http//www.mayyoubehappy.com/unlin
  inev.html The Taniyama-Shimura conjecture. ã
  2001-2004
• 10. OConnor, JJ. http//www-groups.dcs.st-and.ac.
  uk/history/Mathematicians/Dirichlet.html Johann
  Peter Gustav Lejeune Dirichlet. ã 2000