A Brief History of Fermats Last Theorem

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A Brief History of Fermats Last Theorem

• Leo Andrew McCreary Lusk
• Associate Professor of Mathematics
• Gulf Coast Community College

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Pierre de Fermat 1601-1665

• Creator of the worlds most challenging problem.

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Preliminaries

• Fermats Last Theorem is linked with the history
  of mathematics, touching on all the major themes
  of number theory. It provides a glimpse of what
  drives mathematicians and what inspires them.
  The Last Theorem is at the heart of an intriguing
  saga of courage, skullduggery, cunning, and
  tragedy, involving all the greatest heroes of
  mathematics.

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Introduction The Problem

• Although x 2 y 2 z 2 has an infinite number
  of solutions, x n y n z n has no nontrivial
  whole number solutions for n 2.
• For example, we can easily check to see that
  32 42 52 and 52 122 132.
• Try as you may you cannot come up with values for
  x, y, and z such that x 3 y 3 z 3.

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The Story of FLT

• The story of Fermats Last Theorem revolves
  around the search for a missing proof.
• The proof is said to be missing because Fermat
  had claimed he had found one.

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Pythagoras (c. 530 BCE)

• Pythagoras observed that the Egyptians and
  Babylonians conducted each calculation in the
  form of a recipe that could be followed blindly.

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The Right Triangle Relationship

• Pythagoras notes a relationship and subsequent
  recipe that applies to all right triangles. He
  writes, In a right-angled triangle a square
  built upon the hypotenuse is equal to the sum of
  the squares built upon the other two sides.
• In symbols, x 2 y 2 z 2.

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The Right Triangle and the Pythagorean Theoremx
2 y 2 z 2
y
z
x
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Diophantus (c. 250 AD)

• Diophantus, considered the last champion of Greek
  mathematical tradition, writes his now famous
  Arithmetica.
• Of the thirteen original Books of Arithmetica
  only six survived the turmoils of the Dark Ages.
• These Books however would go on to inspire the
  Renaissance mathematicians, including Pierre de
  Fermat.

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Claude Gaspar Bachet (1581-1638)

• Bachet, reputedly the most learned man in all of
  France, realized that Diophantus collection of
  problems found in Arithmetica were on a higher
  plane and worthy of deeper study.

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• Bachet is inspired to make a Latin translation of
  Arithmetica to increase the size of its audience.
• In 1621 when it is published Bachet was
  contributing to the second golden age of
  mathematics.
• As history would later note, it is fortunate that
  he left wide margins in this copy.

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• In 1637, while studying Book II of the new
  Arithmetica, Fermat came across a whole series of
  observations and solutions connected to
  Pythagoras theorem.
• Fermat was struck by the variety and sheer
  quantity of Pythagorean triples and wondered what
  could be added to this subject.

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• In a moment that would immortalize Fermat he
  altered the Pythagorean equation by changing the
  powers to include any positive integer, x n y n
  z n for n 3, 4, 5, . . . And according to
  Fermat there appeared to be no three numbers that
  would perfectly fit this equation.

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• In the wide margins of his copy of Arithmetica,
  next to Problem 8, Fermat made a now famous note
  of his observation

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• Cubem autem in duos cubos, aut
  quadratoquadtradtum in duos quadratoquadratos, et
  generaliter nullam in infinitum ultra quadratum
  potestatem in duos eiusdem nominis fas est
  dividere.

• It is impossible for a cube to be written as a
  sum of two cubes or a fourth power to be written
  as the sum of two fourth powers or, in general,
  for any number which is a power greater than the
  second to be written as a sum of two like powers.

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• The mischievous genius jotted down an additional
  comment that would haunt generations of
  mathematicians

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• Cuius rei demonstrationem mirabilem sane detexi
  hanc marginis exiguitas non caperet.

• I have a truly marvelous demonstration of this
  proposition which this margin is too narrow to
  contain.

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• Fermat had a habit of challenging Europes best
  mathematicians with problems and puzzles, but
  this was Fermat at his most infuriating. While
  his own words suggest that he was pleased with
  this truly marvelous proof, he clearly had no
  intention of writing out the details for anyone
  else.

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• Fermat did however include a sketchy proof for
  the case of n 4 in another section of his copy
  of Arithmetica and used it in the proof of a
  totally different problem. The proof used a newly
  discovered technique called the method of
  infinite descent.

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• January 12, 1665 Fermat died, and still isolated
  from the mathematical community, and not
  necessarily fondly remembered by many of them,
  his discoveries were at risk of being lost
  forever.

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• In 1670, five years after Fermats death, his
  son, who appreciated his fathers mathematical
  hobby collected his notes and letters and had a
  special edition of Arithmetica published as
  Diophantuss Arithmetica Containing Observations
  by P. de Fermat.

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• Fermats many theorems and observations ranged
  from the fundamental to the simply amusing.
• Many mathematicians believed it was essential
  that every single one of Fermats theorems be
  proven - owing to his previous track record.

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• As Europes leading mathematicians band together
  in an all out assault on Fermats theorems they
  fell like dominos in a line, one after the other.
• Until, only one remained.
• This last one would be called Fermats Last
  Theorem.
• FLT would remain unsolved for nearly 350 years
  and would stand as an icon in the area of number
  theory becoming its most valued prize.

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The Pursuit for a Solution

• In August of 1753, 83 years after the special
  publication of Arithmetica, Leonhard Euler
  successfully proved for n 3 that no positive
  whole number solutions exist for FLT.

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• But a century after Fermats death there existed
  proofs for only two specific cases of the Last
  Theorem. Fermats for n 4, and Eulers for n
  3.
• By the beginning of the 19th century, FLT had
  established itself as the most difficult problem
  in the world.
• It seemed like no additional progress could or
  would be made.

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A Dramatic Announcement

• A dramatic announcement by a Frenchwoman was to
  reinvigorate the pursuit of Fermat lost proof.
• Her name is Sophie Germain (1776-1831).

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Sophies Breakthrough

• In the early 1820s Sophie Germain adopted a new
  strategy and described to the great German
  mathematician Karl Friedrich Gauss a so-called
  general approach to the problem.
• Instead of trying to prove particular cases she
  set about describing solutions for many cases all
  at once.

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A Prize is Offered

• After Germains breakthrough the French Academy
  of Sciences offered a series of prizes, including
  a gold medal and 3,000 Francs to the
  mathematician who could finally put to rest the
  mystery of FLT.
• Later, the Paris Academy would offer a prize and
  in 1908, the Wolfskehl prize of 100,000 Marks
  would be offered.

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Solutions Come Forth

• In 1825 Adrien-Marie Legenre succeeded in
  providing a solution for n 5.
• In 1832 Peter Lejeune-Dirichlet proved the
  theorem for n 14.
• In 1839 Gabriel Lamé proved the result for n 7.

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• In 1847 the French Academy of Sciences held one
  its most dramatic meetings when Lamé took the
  floor in front of Europes most eminent
  mathematicians and announced that he had the
  final proof for FLT.
• The stunned audience was setback further when
  Augustus Cauchy asked for permission to speak and
  also told the assembly that he had the final
  proof.

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The Solutions are Flawed

• Lamé and Cauchy each presented their results to
  the Academy but Ernst Kummer would show their
  proofs were flawed.
• In 1850 the Paris Academy offers a prize for the
  solver of FLT.
• It appears though that many mathematicians are
  becoming fearful of trying to solve FLT in light
  of failure.

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Karl Friedrich Gauss 1777-1855

• Gauss, acknowledged as the most brilliant
  mathematician in Europe, is encouraged by
  everyone to compete for the prizes. Says Gauss,
  I confess that FLT is an isolated proposition
  that has very little interest for me.

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Its Beyond Our Capabilities

• In 1857 Kummer demonstrates that a complete proof
  of FLT was beyond the current mathematical
  approaches.
• It was a brilliant piece of mathematical logic,
  but a massive blow to an entire generation of
  mathematicians who had hoped they would solve the
  worlds greatest problem.

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A Challenge

• In 1900 David Hilbert addresses the International
  Congress of Mathematicians in Paris. He
  challenges the mathematical community to solve
  FLT.

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Additional Cases are Solved

• Specific cases of FLT are proven in 1909, 1915,
  and 1953. But still no one is able to come up
  with the complete proof to Fermats Theorem.

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In A Related Matter

• In 1955, in what would become a related matter
  the Taniyama-Shimura conjecture claims that all
  elliptical curves are modular.

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The Last Problem

• In 1963, a 10-year old Andrew Wiles read E.T.
  Bells book The Last Problem. Wiles would be held
  captive by the allure of FLT. It would become his
  own personal passion.

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A Change in Course

• In 1975 Andrew Wiles began his career as a
  graduate student at Cambridge University. His
  major professor is John Coates.
• Coates directs Andrews studies away from things
  like FLT and encourages him to study elliptic
  curves.

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Another Related Matter

• In 1985 the German mathematician Gerhard Frey
  claimed that if the Taniyama-Shimura conjecture
  can be proven then it would follow that FLT is
  true!
• In 1986 Ken Ribet, a professor at Univ. of
  Calif., with the help of Barry Mazur, proves
  Freys conjecture.
• FLT is now linked to the Taniyama-Shimura
  Conjecture.

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Electrified

• Wiles, now a professor at Princeton, learns of
  Ribets breakthrough. Says Wiles, I was
  electrified. I knew at the moment the course of
  my life was changing. All I had to do was prove
  Taniyama-Shimura .

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Secluded Work

• Beginning in the fall of 1986, Wiles abandons all
  work not related to FLT working mostly in the
  privacy of his attic office.
• No other mathematician thinks such a proof can be
  created at this time.

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Struggles

• After five years of secretive and unsuccessful
  work using a technique called Iwasawa theory,
  Wiles resurfaces to see what innovative
  techniques may have been uncovered.
• Wiles attends a conference in Boston on elliptic
  curves and is introduced to a new method of
  analyzing them, the Kolyvagin-Flach method.

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Confides in a Friend

• In 1993 Wiles decides he must confide in someone
  - he chooses his friend and colleague Nick Katz.
  Katz recalls, . . . He (Wiles) said he thought
  he could prove Taniyama-Shimura. I was amazed,
  flabbergasted - this was fantastic!

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The Mock Lecture

• Wiles and Katz needed some way to test his
  conjectures. They set up a course for graduate
  students on Calculations on Elliptic Curves.
• The material was so difficult after a few weeks
  only one person was left in the audience - Katz.

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Eureka?

• Katzs assessment was that the Kolyvagin-Flach
  method was working perfectly. Nobody else in the
  department realized what was about to happen.
  Their plan of secrecy had worked.

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• Once the mock lecture series was over Wiles
  devoted all his efforts to completing the proof.
  Only one family of elliptic curves refused to
  submit to the technique.
• By May of 1993 Wiles was sure he had the whole of
  FLT solved and was preparing to present it at the
  upcoming conference at Cambridge that summer.

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The Lecture

• In June of 1993 Wiles begins his 3-part lecture
  series at Cambridge. The title is Modular Forms,
  Elliptic Curves, and Galois Representations.
  There is no mention of FLT.

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• Wiles first lecture was mundane but was laying
  the foundations for his attack on FLT.
• The astute members of his audience quickly
  realized there was only one direction he could be
  going and the rumors began after he finished his
  first lecture.

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The Second Lecture

• The following day lecture had a significantly
  larger audience. Wiles only teased his audience
  with an intermediate calculation that showed he
  was clearly trying to tackly the Taniyama-Shimura
  conjecture.
• But the audience was still left wondering if he
  had done enough.

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• On June 23 Wiles began his third and final
  lecture. What was truly remarkable was that
  practically everyone who contributed to ideas
  behind the proof was there Mazur, Ribet,
  Kolyvagin, and many others.
• Recalls Mazur, Tension had built for several
  days. There was only one possible punchline.

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I think Ill stop here

• At the final lecture Wiles recalls how many in
  the audience were taking photos.
• The audience was quiet though as Wiles finished
  the proof and he said, I think Ill stop here,
  the audience erupted into sustained applause.

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• It was a completely marvelous event. I mean, you
  go to a conference and there are some routine
  lectures, some good, some special, but its only
  once in a lifetime that you get a lecture where
  someone claims to solve a problem that has
  endured 350 years. We were looking at each other
  saying, My God, weve just witnessed an
  historical event. - Ken Ribet

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A Slight Problem

• Wiles gigantic proof is assigned six referees to
  check its accuracy. He would anxiously await for
  them to complete their task. Nick Katz is one of
  the referees.
• In late August, Katz, Wiles friend in the mock
  lecture series detected a slight problem in the
  calculations.

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To No Avail

• Wiles works feverishly to repair the problem -
  but to no avail.
• By November of 1993 the mathematical community
  was aware of the gap in Wiles proof.
• In December of 1993 Wiles admits to the error and
  continues to work on it. It would not be until
  Sept. 1994 before he could resolve it.

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• Recalls Wiles, I was sitting at my desk
  examining my method. It wasnt that I thought I
  could make it work, but I thought I could at
  least explain why it didnt work. Suddenly, I had
  this incredible revelation. I realized although
  Kolyvagin-Flach wasnt working completely, it was
  all I needed to make my original Iwasawa thoery
  work!

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• A tearful Wiles recalls, It was so indescribably
  beautiful it was so simple and so elegant. I
  couldnt understand how Id missed it and I just
  stared at it in disbelief for twenty minutes.
  During the day I walked around, and Id keep
  coming back to see if it was still there. It was.
  I couldnt contain myself. Nothing I ever do
  again will mean as much.
• Oct. 25, 1994 two papers are released - one was
  Wiles 109 page proof of FLT.

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Wiles and Fermat -Forever Linked

• Wiles was able to prove FLT, that x
  n y n z n has no whole number solutions for
  n 2.

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Resources and References

• Additional info about FLT can be found on the
  Internet.
• An excellent book about FLT Is Simon Singhs The
  Epic Quest to Solve The Worlds Greatest
  Problem.
• An outstanding video is found in the Nova series
  and is called The Proof.