What the Theorem Says
Fermat's Last Theorem states: there are no positive integers x, y, z and integer n > 2 such that xⁿ + yⁿ = zⁿ. For n = 2, there are infinitely many solutions (Pythagorean triples: 3² + 4² = 5², etc.). For n ≥ 3, the theorem claims there are none. Simple to state; extraordinarily difficult to prove.
Three Centuries of Attempts
Euler proved the n = 3 case in 1770. Dirichlet and Legendre proved n = 5 in 1825. Lamé proved n = 7 in 1839. Kummer proved the theorem for all 'regular primes' in 1850. By the 1980s, Fermat's Last Theorem had been verified computationally for all n up to 125,000 — but computation is not proof. The general case resisted everyone.
Wiles's Proof
Andrew Wiles had been obsessed with Fermat since reading it as a 10-year-old. In 1986, Ken Ribet proved the Taniyama-Shimura conjecture implies Fermat's Last Theorem — if all elliptic curves are modular, then FLT is true. Wiles spent 7 years in secret proving the Taniyama-Shimura conjecture for semistable elliptic curves. He announced the proof in June 1993. A gap was found. After 14 months of additional work, the complete proof appeared in 1995 — 130 pages of cutting-edge number theory. The margin was indeed too narrow for Fermat's proof — because his proof almost certainly didn't exist.
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What is Fermat's Last Theorem?
Fermat's Last Theorem states that for integers n > 2, there are no positive integer solutions to xⁿ + yⁿ = zⁿ. For n = 2, solutions are abundant (Pythagorean triples like 3,4,5). Fermat claimed to have a proof in 1637 but never wrote it. Andrew Wiles proved it in 1995 using elliptic curves and modular forms — mathematics that didn't exist in Fermat's era.
Did Fermat actually have a proof?
Almost certainly not. Wiles's proof uses 20th-century mathematics (elliptic curves, modular forms, Galois representations) that Fermat could not have known. The proofs of special cases (n=3, n=5, n=7) already required substantial innovations beyond Fermat's era. The consensus is that Fermat either had a flawed argument he didn't check carefully, or confused himself with a simpler related result.
How did Andrew Wiles prove Fermat's Last Theorem?
Wiles proved the Taniyama-Shimura conjecture for semistable elliptic curves: every such curve corresponds to a modular form. Ken Ribet had earlier proved this implies Fermat's Last Theorem. Wiles worked secretly for 7 years at Princeton, announced a proof in 1993 (which had a gap), then completed the corrected proof in 1994, published in 1995 in 130 pages.