A Brief History of Fermat’s Last Theorem
Pierre de Fermat 1601-1665
Preliminaries
Introduction: The Problem
The Story of FLT
Pythagoras (c. 530 BCE)
The Right Triangle Relationship
The Right Triangle and the Pythagorean Theorem x 2 + y 2 = z 2
Diophantus (c. 250 AD)
Claude Gaspar Bachet (1581-1638)
Bachet is inspired to make a Latin translation of Arithmetica to increase the size of its audience.
In 1637, while studying Book II of the new Arithmetica, Fermat came across a whole series of observations and solutions connected to Pythagoras’ theorem.
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.
In the wide margins of his copy of Arithmetica, next to Problem #8, Fermat made a now famous note of his observation:
“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.”
The mischievous genius jotted down an additional comment that would haunt generations of mathematicians:
“Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet.”
Fermat had a habit of challenging Europe’s 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.
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.”
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.
In 1670, five years after Fermat’s death, his son, who appreciated his fathers mathematical hobby collected his notes and letters and had a special edition of Arithmetica published as “Diophantus’s Arithmetica Containing Observations by P. de Fermat.”
Fermat’s many theorems and observations ranged from the fundamental to the simply amusing.
As Europe’s leading mathematicians band together in an all out assault on Fermat’s theorems they fell like dominos in a line, one after the other.
The Pursuit for a Solution
But a century after Fermat’s death there existed proofs for only two specific cases of the Last Theorem. Fermat’s for n = 4, and Euler’s for n = 3.
A Dramatic Announcement
Sophie’s Breakthrough
$$$ A Prize is Offered $$$
Solutions Come Forth
In 1847 the French Academy of Sciences held one its most dramatic meetings when Lamé took the floor in front of Europe’s most eminent mathematicians and announced that he had the “final proof” for FLT.
The Solutions are Flawed
Karl Friedrich Gauss 1777-1855
It’s Beyond Our Capabilities
A Challenge
Additional Cases are Solved
In A Related Matter
The Last Problem
A Change in Course
Another Related Matter
“Electrified”
Secluded Work
Struggles
Confides in a Friend
The Mock Lecture
Eureka?
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.
The Lecture
Wiles’ first lecture was mundane but was laying the foundations for his attack on FLT.
The Second Lecture
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.
“I think I’ll stop here”
“It was a completely marvelous event. I mean, you go to a conference and there are some routine lectures, some good, some special, but it’s 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, we’ve just witnessed an historical event.’” - Ken Ribet
A Slight Problem
To No Avail
Recalls Wiles, “I was sitting at my desk examining my method. It wasn’t that I thought I could make it work, but I thought I could at least explain why it didn’t work. Suddenly, I had this incredible revelation. I realized although Kolyvagin-Flach wasn’t working completely, it was all I needed to make my original Iwasawa thoery work!”
A tearful Wiles recalls, “It was so indescribably beautiful; it was so simple and so elegant. I couldn’t understand how I’d missed it and I just stared at it in disbelief for twenty minutes. During the day I walked around, and I’d keep coming back to see if it was still there. It was. I couldn’t contain myself. Nothing I ever do again will mean as much.”
Wiles and Fermat - Forever Linked
Resources and References
Email: llusk@ccmail.gc.cc.fl.us
Home Page: http://www.gc.cc.fl.us/mathematics/leolusk.htm
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