Fermat's Last Theorem

Statement of the theorem

For every integer n > 2, the equation

aⁿ + bⁿ = cⁿ

has no solutions in positive integers a, b, c. The exponent n = 2 is the only case with solutions — infinitely many, in fact, given by Pythagorean triples (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), …

Quick exhaustive search for n = 3 — try every (a, b) with 1 ≤ a, b ≤ 50:

a³ + b³ = c³  for c integer?
1³ + 1³ = 2    not a cube
1³ + 2³ = 9    not a cube
2³ + 3³ = 35   not a cube
3³ + 4³ = 91   not a cube
…
no matches found, ever, for n ≥ 3 at any size.

It is enough to prove the theorem for n = 4 and for every odd prime n = p ≥ 3. Reason: if aⁿ + bⁿ = cⁿ for composite n = km, then (a^k)^m + (b^k)^m = (c^k)^m, so a solution at exponent n forces one at every prime divisor of n.

The 357-year story

1637. Fermat reads Diophantus's Arithmetica (Bachet's 1621 Latin edition) and writes in the margin next to Problem II.8 ("split a square into two squares"): Cubum autem in duos cubos … nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere. Cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet. ("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 powers of the same kind. I have discovered a truly marvelous proof which this margin is too narrow to contain.")

1670. Fermat's son Samuel publishes the marginal notes posthumously. The Last Theorem is in print.

~1640. Fermat proves the n = 4 case using "infinite descent," his own technique. The proof survives.

1770. Euler proves n = 3 (with a small gap involving Z[ζ_3] which Legendre and Gauss later filled).

1825. Dirichlet and Legendre independently prove n = 5.

1839. Lamé proves n = 7.

1850. Kummer uses ideal theory in cyclotomic fields to prove FLT for all "regular primes" — primes p such that p does not divide the class number of Q(ζ_p). This handles every prime below 100 except 37, 59, 67. Kummer's irregular primes 37, 59, 67 are handled separately.

1908. Paul Wolfskehl bequeaths 100,000 Marks for a proof — the Wolfskehl Prize. Crackpot submissions flood mathematics departments for the next 90 years.

1955. Yutaka Taniyama conjectures every elliptic curve over Q is associated with a modular form. André Weil and Goro Shimura refine the conjecture in the 1960s — Taniyama–Shimura–Weil.

1984. Gerhard Frey publishes a startling observation: if a^p + b^p = c^p with p prime ≥ 5, the elliptic curve E_F: y² = x(x − a^p)(x + b^p) would have such extraordinary properties that it cannot be modular. So Taniyama–Shimura would imply FLT.

1986. Ken Ribet proves Serre's epsilon conjecture, which makes Frey's link rigorous: a non-modular semistable elliptic curve with the right level structure would imply a contradiction in the theory of modular forms.

1986–1993. Andrew Wiles works in near-total secrecy from his Princeton attic on the modularity conjecture for semistable elliptic curves.

June 23, 1993. Wiles delivers the third of three lectures at the Newton Institute, Cambridge. Slide 21 quietly states modularity, then he adds: "This implies Fermat's Last Theorem. I think I'll stop here." Applause. The story breaks on the front page of The New York Times.

Autumn 1993. Nick Katz, refereeing the manuscript, finds a critical gap in the Euler system argument.

Sept–Oct 1994. Wiles and his former student Richard Taylor switch to a Kolyvagin–Flach approach combined with Hecke-algebra methods. October 1994 the gap closes.

1995. Two papers appear in Annals of Mathematics: Wiles, "Modular elliptic curves and Fermat's Last Theorem" (109 pages); Taylor–Wiles, "Ring-theoretic properties of certain Hecke algebras" (24 pages). FLT is officially proved.

1999. Breuil, Conrad, Diamond, Taylor extend modularity to all elliptic curves over Q — completing Taniyama–Shimura–Weil.

2016. Wiles receives the Abel Prize, the field's most prestigious lifetime honour, citing FLT and modularity.

Outline of the proof

1. Suppose FLT fails. There exist positive integers a, b, c and a prime p ≥ 5 with a^p + b^p = c^p. (The n = 3, 4 cases are already proved.) Without loss assume gcd(a, b, c) = 1.
2. Construct the Frey curve. Define the elliptic curve E_F: y² = x(x − a^p)(x + b^p) over Q. Its discriminant is (abc)^(2p), unusually rich in p-th powers.
3. Ribet's theorem (1986). If E_F were modular, the level-lowering theorem (Ribet) would force a modular form of level 2 to exist. But there are no cusp forms of level 2 and weight 2. Contradiction. So E_F is not modular.
4. Wiles's modularity theorem (1994). Every semistable elliptic curve over Q is modular. The Frey curve E_F is semistable.
5. Contradiction. Steps 3 and 4 together say E_F is both modular and not modular — impossible. The assumption that FLT fails must be wrong.

Wiles's contribution — step 4 — fills 109 dense Annals pages: deformation theory of Galois representations, the R = T theorem identifying a universal deformation ring with a Hecke algebra, and meticulous case analysis depending on the residual mod-p representation. The Taylor–Wiles trick uses auxiliary primes to bootstrap the R = T identification when the original Euler-system approach fails.

Why FLT matters beyond itself

• Modularity theorem. The chief by-product. Wiles's proof handled semistable curves; Breuil–Conrad–Diamond–Taylor (2001) extended to all elliptic curves over Q. This is now the foundational result in arithmetic geometry — every later L-function identity in the area is built on it.
• Langlands program. Modularity is a special case of the Langlands correspondence between Galois representations and automorphic forms. FLT is the test problem that drove the development of the techniques used across the program.
• R = T methodology. Wiles's identification of a deformation ring with a Hecke algebra is now standard practice. Applications include the Sato–Tate conjecture (proved 2008, Clozel–Harris–Shepherd-Barron–Taylor) and Serre's modularity conjecture (Khare–Wintenberger 2009).
• Diophantine geometry. Faltings's 1983 theorem (Mordell conjecture) showed any curve of genus ≥ 2 has only finitely many rational points; the curves x^n + y^n = 1 for n ≥ 4 fall into this class. FLT strengthens "finitely many" to "exactly the trivial ones."
• Public mathematics. Wiles's proof became one of the most-publicised mathematical events of the 20th century. Simon Singh's book (1997) and the Horizon documentary (1996) brought the story to millions.
• Computer-checked proofs. FLT's proof spawned formal verification efforts; Lean 4 and Coq projects have formalised parts of the modularity proof, with an ongoing community goal to formalise the entire chain.

Variants and generalisations

• Beal conjecture (1993). If A^x + B^y = C^z with gcd(A, B, C) = 1 and x, y, z ≥ 3, then A, B, C share a common prime factor. Open. $1M prize from Andrew Beal.
• Fermat–Catalan conjecture. A^p + B^q = C^r with 1/p + 1/q + 1/r < 1 has finitely many primitive solutions. 10 known. Open in general.
• ABC conjecture (Oesterlé–Masser 1985). Implies FLT for all sufficiently large n by a clean inequality on radicals. Mochizuki claims a proof (2012) — controversial and not generally accepted.
• Asymptotic FLT. Proved for large exponents by various methods before Wiles — e.g. Adleman–Heath-Brown 1985 showed the first case (gcd(abc, p) = 1) holds for infinitely many p.
• FLT over number fields. The equation a^n + b^n = c^n over rings of integers in number fields. Solutions can exist — over Q(√−7), Cohn found 2³ + (−1)³ = (√−7)³·… style identities. The "FLT for Q" is the original.
• Faltings's theorem (1983). Every smooth projective curve of genus ≥ 2 over Q has finitely many rational points. The Fermat curve x^n + y^n = 1 has genus (n−1)(n−2)/2 ≥ 2 for n ≥ 4, so Faltings gives finiteness; FLT gives the count = 0 non-trivial.

JavaScript — exhaustive search confirms no small solutions

// Verify Fermat's Last Theorem for small n and small (a, b)
function searchFermat(n, limit) {
  const hits = [];
  for (let a = 1; a <= limit; a++) {
    for (let b = a; b <= limit; b++) {            // b >= a avoids duplicates
      const sum = Math.pow(a, n) + Math.pow(b, n);
      const c = Math.round(Math.pow(sum, 1 / n));
      if (Math.pow(c, n) === sum && c > b) {
        hits.push([a, b, c]);
      }
    }
  }
  return hits;
}

// n = 2: Pythagorean triples — infinitely many
console.log(searchFermat(2, 30).slice(0, 5));
// [[3, 4, 5], [5, 12, 13], [6, 8, 10], [7, 24, 25], [8, 15, 17]]

// n = 3: none
console.log(searchFermat(3, 100));   // []
console.log(searchFermat(3, 1000));  // [] — bigger limit, still empty

// n = 4, 5, 6: none
console.log(searchFermat(4, 200));   // []
console.log(searchFermat(5, 100));   // []
console.log(searchFermat(6, 60));    // []

// Pythagorean parametrisation: m > n > 0 generates ALL primitive triples
function pythagoreanTriple(m, n) {
  return [m*m - n*n, 2*m*n, m*m + n*n];
}
console.log(pythagoreanTriple(2, 1));   // [3, 4, 5]
console.log(pythagoreanTriple(3, 2));   // [5, 12, 13]
console.log(pythagoreanTriple(4, 1));   // [15, 8, 17]

// The Pythagorean parametrisation exists because x² + y² = 1 is a rational curve
// (genus 0). For n >= 3 the curve xⁿ + yⁿ = 1 has positive genus — no rational
// parametrisation, and by Wiles, no non-trivial rational points either.

Common misconceptions

• "Fermat had a proof we just haven't found." Almost certainly false. The proof Wiles found uses mathematics from 1950–1990 (modular forms, elliptic curves, Galois representations). Fermat had none of it. Fermat's "marvellous proof" was almost surely flawed; he later proved n = 4 by infinite descent but never repeated the general claim.
• "FLT is purely number theory." The proof uses complex analysis, algebraic geometry, representation theory, and the deep arithmetic of modular forms. The "number theory" framing is misleading — it's a problem solved by importing all of modern arithmetic geometry.
• "FLT is about a, b, c arbitrary." Standard requires positive integers. The equation aⁿ + bⁿ = cⁿ has trivial solutions if we allow a = 0 or negative numbers (e.g. 1ⁿ + 0ⁿ = 1ⁿ, or in some interpretations 1³ + (−1)³ = 0³, depending on conventions). FLT excludes these by requiring a, b, c > 0.
• "Wiles proved Taniyama–Shimura entirely." Wiles handled the semistable case (sufficient for FLT). The full Taniyama–Shimura–Weil (every elliptic curve over Q is modular) was completed by Breuil, Conrad, Diamond, and Taylor in 1999–2001.
• "The proof is 109 pages." Plus the companion Taylor–Wiles paper, plus Ribet's level-lowering theorem (~70 pages), plus the modular forms theory (centuries of work). The full chain is tens of thousands of pages.
• "FLT has practical applications." Not directly. The methods developed for FLT (modularity, Galois representations) have applications in cryptography (elliptic curve crypto), error-correcting codes, and theoretical physics. The theorem itself is celebrated for its proof, not its uses.