Greatest science books: Fermat’s Last Theorem is still a must-read about a 350-year maths secret
How does Simon Singh’s classic popular science book, Fermat’s Last Theorem, hold up today?
Did you know that the number 26 is rather special? It is the only number that lies directly between a square number (25 or 52) and a cube number (27 or 33). And to be clear, it’s not just that we’ve never found another instance of this square sandwich. We know for sure that there is no other between zero and infinity.
Simon Singh’s 1997 book Fermat’s Last Theorem is an exploration of mathematical proof – what it means, how it is obtained, and what motivates those who pursue it so passionately. It tells the story of the quest for a particularly tantalizing piece of evidence, making it a compelling read. But given that this proof took 350 years to emerge, it also turns out to be a wonderful history of mathematics. For many of us, the essence of mathematics lies in a realm of abstract reasoning far beyond ourselves. But for me, what makes this book an absolute treasure, even almost 30 years after Singh wrote it, is the way it transports us into the heart of this alluring world.
Singh starts right at the beginning with Pythagoras, famous in the field of triangles. Everyone has heard of the Pythagorean theorem, which says that if you add the squares of the lengths of the two shorter sides of a right triangle, they equal the square of the length of the longer side (an idea that can be expressed as: x2 + yes2 = z2). Others had already used this method to work with triangles, but what distinguished Pythagoras, Singh writes, was that he had proven this to be true for all right triangles. He did this not by trial and error or experiment, but by using unquestionable logic. “The search for a mathematical proof,” writes Singh, “is the search for more absolute knowledge than [that] accumulated by any other discipline.
The story of Pythagoras was actually one of my favorite parts of the book. I didn’t realize he was the founder of a secret brotherhood of evidence seekers. And I read with wide eyes how a man named Cyclon was refused entry into the brotherhood and conspired to have Pythagoras killed in revenge.
But the man who kicks off the story is Pierre de Fermat. He was a judge who lived in France in the first half of the 17th century – and a prodigious mathematical talent. One thing he proved was the uniqueness of the number 26. What made him famous, however, was his so-called Last Theorem, which amounts to a simple extension of the Pythagorean Theorem. We know that there is an infinite range of integers that can be successfully fit into the standard Pythagorean equation, but Fermat assumed that if you fit the equation to xn + yesn = znwhere n can be any integer, then there are no integer solutions. Around 1637, he insolently claimed to have “truly marvelous” proof – but did not write it down.
Imagine 350 years of mathematicians driving themselves half crazy trying to discover the secret. Singh guides us through it all with style and ease, incorporating an incredible cast of characters along the way. Among my favorites were Sophie Germaine, the French mathematician who worked in secret under a man’s name; Évariste Galois, the angry revolutionary who made a giant breakthrough in mathematics, then promptly got himself killed in a duel; and Yutaka Taniyama, the brilliant young Japanese mathematician who helped lay the foundation for the definitive proof of Fermat’s conjecture and then tragically committed suicide.
The main star of our story, however, is mathematician Andrew Wiles, who (spoiler alert) ultimately proves Fermat’s theorem to be true in 1994. Singh paints a wonderfully rich picture of Wiles, which is all the more impressive given that Wiles clearly doesn’t like being in the spotlight. While reading, I had the illusion of understanding roughly what Wiles was doing. In short, it was about building a logical bridge between a branch of mathematics called elliptic curves and another branch called modular forms, which had previously been thought to be chalk and cheese. To say more here would be impossible – it’s obscure, if fascinating, stuff.
There is, however, a tense coda to the story, namely that Wiles’ original proof contained an error. It’s the nightmare scenario, but – perfectly – Wiles rises from the ashes to finally fix the flaw. My only slight criticism of the book would be that this restorative part of the story could have been shorter.
Singh’s book has aged well and its themes remain relevant to modern mathematics. One of the ideas behind both the book and Wiles’ proof is the so-called Langlands program, which originated with mathematician Robert Langlands in 1967. He theorized that, deep down, all areas of mathematics are related. The hope is that by finding these connections, intractable problems in one area of mathematics will suddenly disappear while an arsenal of tools from another area can suddenly be used against them. Wiles’ work was an early clue that the Langlands program might be on the right track – and more have emerged recently. In 2024, mathematicians presented a proof of an aspect of the Langlands conjecture related to an area of mathematics called harmonic analysis.
When I finished the book and put it down, I couldn’t help but feel like I was walking through a gallery full of abstract art. Mathematical proofs are a bit like art, I think. You observe them silently, wondering how the wizards who created them managed to achieve this, and you come away feeling like you’ve glimpsed something beyond the surface of everyday experience. For successfully creating such a feeling, I can only congratulate this book.
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