Solving a 358 years old problem
On 23 June 1993, Andrew Wiles, after 7 long years of private work, during a lecture at Isaac Newton Institute in Cambridge, presented his proof that solved one of the holy grails of mathematics: Fermat Last Theorem (later after peer review, a mistake was discovered in the initial proof, that was fixed next year in September 1994).
Fermat’s last theorem
Fermat’s last theorem, also called Fermat’s great theorem, states that there are no positive numbers x, y, and z such that
xn + yn = zn, for any integer n >2.
For example, if n = 3, Fermat’s last theorem states that no natural numbers x, y, and z exist such that x3 + y 3 = z3 (i.e. the sum of two cubes is not a cube).
Pierre de Fermat was a 17th-century judge and his main hobby was mathematics. In fact, he is mostly remembered for his genius contribution in mathematics rather than for his work as a judge. One evening while reading Arithmetica by Diophantus (one of the books that survived from the Library of Alexandria) he stumbled upon a problem that had its origin in Greek mathematics, the Pythagorean equation x2 + y2 = z2. Fermat’s Last Theorem replaces the squared numbers with any power higher than 2 and raises the question if whole numbers can be found to satisfy this equation. Fermat scrambled on the margins of his book that he knows how to solve this equation: “I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.” And then he dies and his magic proof (“demonstrationem mirabilem”) dies with him and an open challenge to prove it was set for the rest of the world.
Proofs for specific cases
Although Fermat never wrote down a complete proof, he knew how to handle the special case of n = 4 (he sketched a solution on the pages of the same book, while solving another problem). In fact, Fermat’s book had a lot of notes on the margins with claims that he could solve other complicated mathematical problems.
After his death, his son published a new version of the book with Fermat’s notes included in the text. Mathematicians started to challenge themselves to prove all of Fermat’s mathematical claims. Fermat’s Last Theorem is called the last one, because all the other problems he wrote on the book were proved one by one and this one was the last one to be elucidated.
In 1753, Leonard Euler, using an approach similar to the one used by Fermat, managed to prove the case for n = 3. After a hundred years, Euler was the first one to make progress on Fermat’s Last Theorem. This proof is very important due the fact that 3 is a prime number and if one could find a proof for all the prime values of n, that proof could be extended to all numbers. Another important progress was made by Sophie Germain, who was able to find a proof for an entire category of primes (Germain primes).
One great mathematician after another was struggling to crack this problem, performing small steps to finding a complete solution, and for three hundred years, nobody was able to find it.
Andrew Wiles
The story has a happy and amazing ending and it all starts with the 10 years old Andrew Wiles when he found a book called “The last problem” by Eric Temple Bell. This book is about Fermat’s Last Theorem. The problem is a simple, clear statement that could easily be understood by a 10 years old boy fascinated by puzzles: “What amazed me was that there were some unsolved problems that someone who was 10 years old could understand and even try.” – Andrew Wiles.
Sir Andrew Wiles, was awarded the Abel prize in 2016.
Photograph: Charles Rex Arbogast/AP
Andrew Wiles earned a PhD from Cambridge on elliptic curves. These are equations that have this form y2 = x3 + ax2 + bx +c where a,b,c are any whole numbers and the challenge is to find out if these elliptic equations have whole number solutions and how many (in a way similar to Fermat’s Last Theorem).
In 1995, Goro Shimura and Yutaka Taniyama discovered (but not proved) a link between elliptic curves and modular forms: Taniyama–Shimura conjecture. Later on, other mathematicians found a relation between the modularity conjecture and Fermat’s Last Theorem and demonstrated that if someone would prove Taniyama–Shimura conjecture, Fermat’s Last Theorem would be demonstrated as a consequence. This is the entry point that Andrew Wiles needed to attack Fermat’s Last theorem.
The proof for Fermat’s last Theorem is based on the discoveries of great mathematicians and it is an aggregation of tremendous work from multiple mathematical fields. With this proof, a lot of other mathematical conjectures were proved as well (they based themselves on the assumption Taniyama–Shimura conjecture was true).
While being reviewed, a critical error was found in Andrew Wiles proof. Even with the error, the work was amazing and worthy of highest praise but with this gap there was no actual proof of Fermat’s Last Theorem.
The ordeal
For more than one year, Andrew Wiles, working with his former student Richars Taylor, tried to repair the demonstration. Here comes my favorite part: Andrew Wiles was about to give up and accept that he had failed, but at least he wanted to understand why he had failed and understand why the method he used did not work. His own review process triggered an incredible revelation and he came up with a new approach to complete his work. The final proof appeared in the Annals of Mathematics in May 1995 and so the 358 Fermat’s conjecture was finally resolved.
Fermat and Homer Simpson
Thanks to Andrew Wiles, we know for sure that there are no solutions to Fermat’s Last Theorem but in Springfield everything is possible.
Homer Simpson disproves Fermat’s last theorem, by finding an example that fits the criteria. You can easily check Homer’s values on your desktop calculator: 398712 + 436512 = 447212 and you will be amazed by the result.
This is called a “near miss solution” and the error can be spotted only on calculators with a higher decimal precision (although looking at the parity of the numbers you can also spot the mismatch).
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