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Fermat's Last Theorem: the Story of a Riddle That Confounded the World's Greatest Minds for 358 Years by Simon Singh
Kirk A.’s recommendation for the 510 Club | December 2025
In about 1637 Pierre de Fermat scribbled in a copy of Diophantus' Arithmetica a tantalizing challenge to future mathematicians: 'I have a truly marvellous proof of this proposition which this margin is too narrow to contain.' The proposition in question being that the equation an + bn = cn has no integer solutions for n > 2. Singh does an admirable job of bringing to life the millennia old story of one of the longest surviving puzzles in mathematics, starting from the work of the Pythagorean Brotherhood and later Euclid, who demonstrated how to construct an infinite variety of Pythagorean triples (the solutions for n = 2), to the popularization of the so-called last theorem in the 1670s, the partial progress made over the centuries since and finally the nail-biting story of Andrew Wiles' proof, finally confirmed in late 1994 after over a year-long wait after the initial announcement to fix a significant error.
Fermat was famously secretive and reluctant to write down full solutions and proofs. Elsewhere in his copy of Arithmetica he gave an uncharacteristically more complete sketch of a proof that there were no solutions for n = 4. The question of why he would do this if he still had faith in his supposed general solution remains. In 1753 Euler was able to extend Fermat's proof for n = 4 to the case n = 3 using complex numbers. Sophie Germain made significant progress in the early 19th century, confirming whole classes of exponents. This led to a proof by Ernst Kummer in the mid-19th century that the theorem was true for regular primes, leaving just the so-called irregular primes such as 37 and 59 to be considered (these forming approximately 40% of all prime numbers). During the 20th century, increasing access to machine calculating capacity led to numerical confirmation of specific exponents. When news first broke of Wiles' proof in 1993, all exponents less than 4 million had been confirmed.
In post-war Japan the mathematical research community was somewhat isolated and came to study areas of mathematics considered unfashionable elsewhere, one of which was modular forms. From this research mathematics was granted the Taniyama-Shimura conjecture: that for each of the infinite modular forms there was a corresponding elliptical equation, one in the form y2 = x3 + px2 + qx + r. This unexpectedly linked two widely separated branches of mathematics, allowing techniques from one to be made use of in the other. As the truth of the Taniyama-Shimura conjecture became largely accepted (but still unproven), many results became predicated on first assuming it were true. Around this time, Andrew Wiles, who first became inspired by the theorem at the age of 10, was luckily assigned the respectable area of elliptic curves to research when he later entered academia. Seemingly nothing to do with Fermat's Last Theorem!
At a 1984 conference in the Black Forest, Germany, a transformation turning Fermat's equation into an elliptic equation was demonstrated by German mathematician Gerhard Frey. An outline of a method to prove Fermat's Last Theorem was then presented. The first missing step took longer to establish than expected: show that this elliptic equation would be so eccentric that it could not possibly be equivalent to a modular form. This was proved by Ken Ribet, a professor at UC Berkeley in 1986. The second missing step: complete the proof of the Taniyama-Shimura conjecture so that the modular form would nonetheless have to exist.
In a similar fashion to the secretive way Fermat had worked, Wiles began an obsessive sole effort over the course of some seven years to complete the proof, still fulfilling his lecture work but neglecting the conference circuit and occasionally publishing a minor decoy paper to make it look like he was working on some unrelated research. It is uncomfortable to imagine what could have happened if an accident had occurred, perhaps he too, like Fermat, was not making notes that would be understandable by anybody else attempting to carry on from his research. In fact, during the time following the announcement when the publication delay indicated that there was a gap in the proof, some thought that Wiles should publish anyway, so that the world would gain from the incomplete proof, which was known to still contain some significant new mathematics. Wiles' proof, much more than just confirming an isolated fact in number theory, had drawn on and perhaps revitalized many areas of modern mathematical research.
If you haven't already read this history, then I wholeheartedly recommend it to you. Hopefully it won't spoil your daydreams (if you're anything like me) of proving Goldbach's conjecture or that all perfect numbers are even. Perhaps it will inspire you to greater efforts instead!
WHAT IS THE 510 CLUB?
The 510 Club is named after the Dewey Decimal classification for Mathematics. It is a book recommendation project facilitated by Mathateca in collaboration with Christchurch MathsJam. Each month we feature a mathematical book recommendation, whether that’s a novel, articles / essays, a puzzle book, textbook, biography... just as long as it features maths in some way. Read the above book at your leisure then feel free to comment your thoughts below, or come along to the following Christchurch MathsJam sessions to join in an informal maths/book chat with the reviewer.
We're always looking for suggestions! If you're interested in contributing a book rec one month, please email christchurch@mathsjam.com to sign up.
