In 1637, Pierre de Fermat wrote in the margin of his copy of Diophantus’s Arithmetica:

It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain.

In modern notation: the equation

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

has no positive integer solutions for $n > 2$.

For $n = 2$, there are infinitely many solutions — the Pythagorean triples ($3^2 + 4^2 = 5^2$, etc.). Fermat claimed that for every higher power, there are none at all. He almost certainly did not have a correct proof. It would take 358 years and the deepest mathematics of the 20th century to prove him right.

Centuries of partial progress

Fermat himself proved the case $n = 4$ using infinite descent — if a solution exists, you can always construct a smaller one, which is impossible.

Progress came slowly: - Euler (1770): proved $n = 3$ - Dirichlet and Legendre (1825): proved $n = 5$ - Lamé (1839): proved $n = 7$ - Dirichlet (1832): proved $n = 14$

Sophie Germain (early 1800s) proved a general result: if $p$ is an odd prime and $2p + 1$ is also prime (a “Sophie Germain prime”), then there are no solutions with $n = p$ where none of $x, y, z$ is divisible by $p$. This covered infinitely many cases at once.

Kummer (1850s) made the deepest pre-modern progress by introducing ideal numbers — a concept invented specifically to address the failure of unique factorisation in certain number rings. He proved FLT for all regular primes (primes $p$ that don’t divide any of the numerators of the Bernoulli numbers $B_2, B_4, \ldots, B_{p-3}$). The first irregular prime is 37.

By the late 20th century, computers had verified FLT for all $n$ up to about 4 million. But case-by-case verification could never finish — a proof for all $n$ required entirely different ideas.

The connection to unique factorisation

A naive approach to FLT would be to factorise $x^n + y^n$ in $\mathbb{Z}[\zeta_n]$, the integers extended with an $n$-th root of unity $\zeta_n = e^{2\pi i/n}$. For example, $x^3 + y^3 = (x + y)(x + \zeta_3 y)(x + \zeta_3^2 y)$.

If unique factorisation held in $\mathbb{Z}[\zeta_n]$, you could derive a contradiction. But it doesn’t — unique factorisation fails for $n \geq 23$. This is exactly what Kummer discovered, and why he invented ideal numbers (later formalised as ideals by Dedekind).

The failure of the Fundamental Theorem of Arithmetic in extended number rings is what makes FLT hard. In $\mathbb{Z}$, every number factors uniquely into primes. In $\mathbb{Z}[\sqrt{-5}]$, we already saw that $6 = 2 \times 3 = (1+\sqrt{-5})(1-\sqrt{-5})$. FLT lives in this territory — where the algebra we take for granted breaks down.

The modern proof

The proof that finally settled FLT came from an entirely unexpected direction: the theory of elliptic curves and modular forms.

The Frey curve (1985): Gerhard Frey observed that if $a^p + b^p = c^p$ were a solution, the elliptic curve $y^2 = x(x-a^p)(x+b^p)$ would have bizarre properties.

Ribet’s theorem (1986): Kenneth Ribet proved that Frey curves cannot be modular — their properties are too pathological.

Wiles’ theorem (1995): Andrew Wiles proved that all semistable elliptic curves are modular (the semistable case of the Taniyama-Shimura conjecture).

Since Frey curves are semistable, Wiles’ theorem says they must be modular. But Ribet’s theorem says they can’t be modular. Contradiction — so no solution to Fermat’s equation exists.

What the proof revealed

Wiles didn’t prove FLT by studying $x^n + y^n = z^n$ directly. He proved something far more general — a deep structural theorem about the relationship between elliptic curves and modular forms — and FLT fell out as a corollary.

The proof connects: - Number theory: primes, Galois groups, ideal class groups - Algebraic geometry: elliptic curves, Frey curves - Analysis: modular forms, L-functions, Hecke operators - Representation theory: Galois representations, deformation theory

No single subfield of mathematics could have produced this proof. It required seeing connections across domains that had developed independently for centuries. In this sense, FLT is less a theorem about Diophantine equations and more a demonstration that number theory, geometry, and analysis are aspects of a single deeper structure.

Wiles announced his proof on June 23, 1993, at a lecture in Cambridge. The audience, realising what had been achieved, broke into applause. When asked how it felt, he said: “There’s no other problem that will mean the same to me. This was my childhood dream.”