# Simon Singh - Fermat's Last Theorem

**Simon Singh** 2005. *Fermat’s Last Theorem* London: Harper Collins. 340 pp.

Not just a popular account of the efforts of multiple mathematicians to solve Pierre de Fermat’s Last Theorem, but, and for me primarily, a splendid introduction to the rarified atmosphere of mathematicians working on number theory.

Fermat’s Last Theorem was more properly a conjecture, since from the time Fermat stated it in 1637 as a marginal annotation in an older text, and for the next 358 years, it remained unproven. The Theorem states:

there are no whole number solutions for the equation:

x^{n}+y^{n}=x^{n}where n is any number greater than 2

(If n=2 then it becomes Pythagoras’ Theorem which describes the relationship between the sides of a right-angled triangle: *x*^{2} + *y*^{2} = *z*^{2}.)

Fermat’s marginal annotation stated (in Latin)

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

But if he did have a “truly marvellous demonstration” it died with him. Mathematicians ever since have been striving to discover the lost proof.

Thus begins the highly readable account by Simon Singh of the long journey to a proof, from which I took the following messages:

- Mathematics is not science and the two bodies of knowledge are fundamentally different. Mathematics proceeds by developing (discovering) proofs, which once established are universally and eternally true and which typically form the basis of additional new proofs. Scientific knowledge, in contrast, is based on theories that scientists attempt to refute (disprove) by obtaining new, independent data (new observations or new experimental results). If a theory survives multiple attempts to disprove it then it becomes accepted, however it is never safe from new data that may potentially disprove it.
- Number theory - the study of the property of numbers - began in the 6th century BC with the work of Pythagoras of Samos and his school of disciples. Their curiosity and fascination with the properties of numbers was an early example of humanity’s fascination with esoteric and beautiful concepts. Pythagoras and his group (about 600 of them!) were able to follow their fancy in this esoteric field which was then of no utility to humans. Pythagoras was also the first to coin the word
*philosopher*. Thus did humanity begin, or at least accelerate, a long ascent. Pulling ourselves up by our bootstraps. - Others who attempted to tackle Fermat included several whose contributions form the early chapters in the book. Leonhard Euler, (1707 - 1783), the most brilliant and prolific mathematician of his day (his works are still being translated and published) provided a proof for the special case
*x*^{3}+*y*^{3}=*z*^{3}(Fermat himself had published a proof by contradiction for another special case*x*^{4}+*y*^{4}=*z*^{4}.) French mathematician and physicist Sophie Germain (1776 – 1831) was a rare example of a female mathematician who had to pretend to be male to get her initial schooling. Eventually her brilliance was recognised and sponsored by other notable mathematicians of her day, David Hilbert and Carl Friedrich Gauss. Germain’s approach was more general than preceding workers and focused on a particular class of prime numbers*p*such that (2*p*+ 1) was also prime. Singh discusses several other European mathematicians of the 1800s and 1900s who made further incremental advances, before allowing himself what seems something of a digression discussing the brilliant but disturbed hypochondriac logician Kurt Gödel and even Alan Turing. - Two Japanese mathematicians Yutaka Taniyama and Goro Shimura made a highly technical contribution that eventually was crucial to solving Fermat’s Theorem. They stated what is known (among several other attributions) as the Taniyama–Shimura conjecture. Taniyama and Shimura were fascinated by the famously (among mathematicians) wierd and wonderful
*modular forms*which are mathematical objects remarkable for their infinite symmetry. Taniyama and Shimura conjectured that there was a relationship between*modular forms*and another esoteric mathematical object called*elliptic equations*(or*elliptic curves*). Singh provides a virtuoso description of these highly technical concepts. Taniyama tragically and inexplicably committed suicide in 1958, yet another example of the personal tragedy of brilliant but flawed mathematicians throughout history, several being part of this book. - Cambridge- and Princeton-based mathematician Andrew Wiles was driven to work on Fermat’s Theorem since childhood and spent 7 years in solitary work in the attempt. Wiles’ sustained solitary effort was unique in modern mathematics where collaboration through conference presentations and preprint publication on the web is the norm. However Wiles’ most momentous contribution (to me) was bringing together two seemingly unrelated fields of mathematics: he provided a proof for a significant case of the Taniyama–Shimura conjecture and showed that this was sufficient to also imply that Fermat’s Last Theorem was also proven. However his work contained a flaw that took him and Cambridge colleague Richard Taylor a further year to remedy (Wiles by then realised he could not achieve his goal alone). The linking of two seemingly unrelated fields is a recurring theme in mathematical and scientific breakthroughs and is arguably the true mark of genius.
- Given that none of the mathematical tools used by Wiles and colleagues were available to Fermat, and that the eventual proof ran to over a hundred highly technical pages, the obvious question arises: Did Fermat really have a proof, as he stated but never published? Singh spends a very few concluding words on this, without committing to an opinion. Either Fermat was wrong, or, tantalisingly, perhaps another simpler proof using 17th century mathemtaical techniques exists and awaits rediscovery. It is nice, but probably unrealistic, to think so.