The problem
For n > 2, the equation xⁿ + yⁿ = zⁿ has no solutions in positive integers x, y, z.
Fermat's Last Theorem
historicReferences
History & significance
Written by Fermat in his copy of Diophantus around 1637 with the fatal words "I have discovered a truly marvellous proof of this, which this margin is too narrow to contain." Euler settled n = 3; Dirichlet and Legendre n = 5; Kummer built ideal theory reaching all regular exponents. Sophie Germain's partial programme anticipated modern approaches.
Connected problems
The resolution (human proof)
Proved by Andrew Wiles — announced June 1993, repaired with Richard Taylor, published 1995. The proof proves the Taniyama–Shimura modularity conjecture for semistable elliptic curves: FLT follows because a counterexample would yield a non-modular semistable curve (Frey's 1985 observation, Ribet's 1986 theorem). Seven years of secret work; the 1993 gap took another eighteen months. The archetypal century-scale human conquest — and the template for how a single mind with the right structures can end a 358-year story.