Published in 1997 as a follow up to a BBC documentary about the discovery of a proof of Fermat’s last theorem in 1994/5 this 362 page book takes a deep dive into the history of the theorem and the various attempts at a solution over the 358 years that it remained a mathematical puzzle. The joy of Pierre de Fermat’s last theorem is that it is very simple to understand but turned out to be incredibly difficult to prove. Anyone who has had Pythagoras’s theorem relating to the sides of a right angled triangle drummed into them at school will understand the basic concept. That theorem states that the square on the hypotenuse equals the sum of the squares on the other two sides or put simply in a diagram with the best known whole number solution as an example:

Fermat stated that although this clearly works for squared numbers, and indeed there are infinitely more combinations of whole number solutions (such as x = 5, y=12 and z=13 as 25 + 144 = 169) there are no whole number solutions if the power that you raise x, y and z to is greater than 2. Fermat stated that he had a proof, although he wrote this in the margin of his copy of the ancient Greek mathematician Diophantus’s Arithmetica and stated that ‘the proof was too large to fit in the margin’. Fermat was a mathematical genius but also extremely annoying as he would often taunt fellow mathematicians by writing to them that he had discovered a proof to some mathematical conjecture and challenge them to also find the solution and would rarely write down his own proofs in a rigorous manner. Certainly no example of Fermat proving his last theorem has ever been found. Fermat of course didn’t refer to it as his last theorem, it gained the name as slowly all his other conjectures were proved correct leaving just this one which would become notorious and also the driver of other mathematical insights as people tried to prove, or disprove, it over more than three and half centuries.

Let’s come back to that date of 1994/5 for the final proof. English mathematician Andrew Wiles had worked for many years on attempting a proof, but without admitting to his fellow mathematicians that he was working on it as it was seen as a waste of time and as a professor at Princeton University, New Jersey, USA it wouldn’t be appropriate to be seen to have an interest in the subject. However all that changed in the mid 1980’s when it was shown that Fermat’s last theorem would be effectively proved if there was a proof discovered to the seemingly unrelated Taniyama-Shimura conjecture. This conjecture deals with two extremely complex areas of mathematics and indicated that they were inter-related and indeed one could be used to solve problems in the other. These two concepts were elliptic curves, which were Wiles’s Phd speciality and modular forms, a four dimensional topological ‘structure’. Now I sort of understand the basics of elliptic curves but the use of modular forms is beyond me even with the basic description provided by Simon Singh in this book. Wiles saw this as a legitimate use of his time and would give him a proof of Fermat, which had fascinated him since he was ten years old, whilst working on a ‘genuine’ mathematical problem, the proof of Taniyama-Shimura. The problem was that this, like Fermat’s last theorem, was considered impossible to prove. He still decided to work in secret though and for many years came up against brick walls preventing his proof from working until in 1994 he took three lecture slots at a convention in Cambridge, England and under the deliberately opaque title of “Modular Forms, Elliptic Curves and Galois Representations” endeavoured to present his proof. The mathematical world was astounded and Wiles was hailed for his outstanding achievement, problems however were found during the rigorous checking before the proof could be published and it took several more months before Wiles finally fixed the error in his proof hence 1994/5 being given as the date of the solution. The 1994 proof was so close to being correct, but relied on another conjecture which it turned out wasn’t proved so proving this other theorem was what took the extra time.

Now it may well be, if you are still reading this blog, that you are thinking no way am I going to read this book it sounds far to complex but you would be wrong. Singh has done a remarkable job in not only summarising Andrew Wiles’s work and still making it approachable, but the history of the various attempts to solve Fermat is fascinating. I first read this book back in 1997 when it came out and have picked it off the shelves two or three times in the intervening decades and each time I love descriptions of the failed attempts and the progress, or otherwise, that they led to, along with the various other puzzles included which help to get your brain engaged in the problem. Each time I get that little bit further in understanding just what Wiles actually proved with the specific part of the Taniyama-Shimura conjecture (named after the two Japanese mathematicians who came up with it in 1957). Taniyama-Shimura would finally be proved for all variants in 2001 by four of Andrew Wiles’s former students and renamed The Modularity Theorem. A note on terminology conjectures are unproved but seem to work, theorems are fully proved

Give your brain a workout, I definitely recommend giving it a go.