# Fermat’s Last Theorem

I have been re-reading Simon Singh‘s excellent Fermat’s Last Theorem, a biography of the famous mathematical theorem (although for the past 350 years it should more accurately have been referred to as Fermat’s Last Conjecture):

or:

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:

$x^n + y^n = z^n$

Pierre de Fermat was a French lawyer whose hobby was mathematics, a true amateur academic. Fermat is often referred to as the “Prince of Amateurs”, but his contribution to mathematics during the 17th century was so great that he should be counted as a professional mathematician.

Fermat was also well known for steadfastly refusing to reveal his proofs — publication and recognition meant nothing to him as he was satisfied with the simple pleasure of being able to solve problem in mathematics. However, this shy and retiring genius also had a mischievous streak, which, when combined with his secrecy, meant that when he did communicate with other mathematicians, it was only to tease them. He would write letters stating his most recent theorems without providing the accompanying proof, whilst also challenging his contemporaries to find the proof. Descartes called Fermat a “braggart“; English mathematician John Wallis referred to him as “That damned Frenchman“. Pascal urged Fermat to publish his work, who replied: “Whatever of my work is judged worthy of publication, I do not want my name to appear there.

Much of Fermat’s mathematical inspiration came from his copy of DiophantusArithmetica, a collection of 130 algebraic problems giving numerical solutions of indeterminate equations (Diophantine equations). In was in the margin of his Arithmetica, next to Problem VIII, that he made his famous observation:

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

which means:

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

This was Fermat at his most frustrating. While he left no proof of the conjecture for all $n$, he did prove the special case $n=4$. As his own words suggest, he was particularly pleased with this “truly marvellous” proof, but he had no intention of writing out the detail of the argument, let alone publishing it.

Further proofs for specific exponents were contributed by a number of mathematicians, including Euler, Legendre and Hilbert. The problem was reduced to proving the conjecture for exponents $n$ that were prime; Germain proved a special case for all primes less than 100, while Kummer proved it for all regular primes. Building on Kummer’s work, other mathematicians were able to prove the conjecture for all odd primes up to four million, but a general proof appeared to be out of reach.

The final proof of the conjecture for all $n$ came in the late 20th century. Andrew Wiles, building on the work of Gerhard Frey and Ken Ribet, presented a proof of the modularity theorem for semistable elliptic curves (via proof of the Taniyama-Shimura conjecture), applying techniques from algebraic geometry and number theory. His two manuscripts, published in 1995 in the Annals of Mathematics, were the last step in proving Fermat’s Last Theorem, 358 years after it was conjectured: “Modular elliptic curves and Fermat’s Last Theorem” and “Ring theoretic properties of certain Hecke algebras“. The proof itself is over 100 pages long and consumed seven years of Wiles’ research time.

But the question remains: did Fermat possess a general proof? Wiles’ proof relies on mathematical techniques developed in the 20th century, which would have been alien to Fermat. Most mathematicians and science historians doubt that Fermat had a valid proof of his theorem for all exponents $n$, as it would have had to have been elementary, given mathematical knowledge of the time.

Over three centuries of effort on this problem, enhancing its notoriety as the most demanding riddle in mathematics (even transcending popular culture), all because of small margins.

1. Dave Donaghy says:

How many other problems have such a simple statement, but have commanded such an extraordinary amount of effort?

I know the answer to this must be known – and to my shame, I haven’t the book – but I wonder how many different purported solutions have been presented? I think I remember reading that there was a US President in the list of claimants.

(Oh, and can I borrow the book?)

2. I agree — maybe only $E = mc^2$ for pervading the public’s consciousness (though perhaps with less understanding of what the equation encapsulates)?

It was James Garfield, 20th President of the United States: see Proof #5!

I’ll bring the book over when I’m next in Bath.

3. Cliff Bott says:

I seem to recall reading that Fermat never claimed publicly to have had a proof of the ‘Last Theorem’. The marginal note was published by Fermat’s son when he published Fermat’s collected work. Fermat’s ‘challenges’ contained in letters to other mathematicians had, I believe, all been verified to his satisfaction. Presumably the ‘Last Theorem’ was not among these because Fermat realised later that there was a flaw in his reasoning, but he saw no reason to correct the marginal comment, because it was never intended to become public. Surely if he had had a proof, he would have made the fact public. The idea that Fermat did had a proof that has eluded other mathematicians for 350 years makes for entertaining reading, but is not the most plausible explanation of events.

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