In 1983, Ribet noted that a counterexample to it would imply the existence of a non-modular elliptic curve, contradicting a special case of the Taniyama-Shimura conjecture.
Kummer showed that early attempts at proof of this depended on the faulty assumption that factorization is unique in all rings of integers or algebraic number fields. The correct argument uses L-functions, which are generalized zeta functions; a combination of Kolyvagin-Flach theory and Iwasawa theory; and Hecke algebras. In 1983, Ribet noted that a counterexample to it would imply the existence of a non-modular elliptic curve, contradicting a special case of the Taniyama-Shimura conjecture. In 1995, Taylor and Wiles proved that special case of that conjecture and thus proved, FTP, what centuries-old math problem named for a French margin-scribbler?
Kummer showed that early attempts at proof of this depended on the faulty assumption that factorization is unique in all rings of integers or algebraic number fields.
Fermat's last theorem