What the work claims
Fermat’s Last Theorem has no integer solutions for n > 2. This follows from the modularity of semistable elliptic curves — a property Wiles proved — combined with Ribet’s theorem, which showed any Fermat counterexample would generate a semistable elliptic curve violating modularity.
How it was done
Wiles worked secretly for six years. He proved the modularity theorem for semistable elliptic curves. He used proof by contradiction: assuming a counterexample to Fermat’s Last Theorem existed, he derived a non-modular semistable elliptic curve — contradicting the modularity theorem. Ribet’s theorem linked Fermat solutions to such non-modular curves. His 1993 Cambridge presentation was followed by peer-review discovery of a flaw in August 1993, concerning the Selmer group and Euler systems. With Richard Taylor, he circumvented it using Galois representations, Kolyvagin’s and Flach’s ideas, and Iwasawa theory.
What holds up
The 1995 proof holds up. It establishes the modularity theorem for semistable elliptic curves. It relies on Ribet’s theorem to link Fermat solutions to non-modular curves. The joint Wiles–Taylor paper validates the corrected steps. The Annals of Mathematics publication confirms formal peer-reviewed acceptance.
What does not
It does not prove the full Taniyama–Shimura conjecture. It does not establish Fermat’s Last Theorem via elementary methods. It does not resolve the general modularity conjecture for all elliptic curves — only the semistable case. It does not eliminate alternative proofs or imply uniqueness of approach.
Why it matters beyond the lab
It matters because it closed the longest-standing open problem in mathematics — not by invention of new arithmetic, but by proving that two distant domains (elliptic curves and modular forms) are inseparable in a critical class. It confirmed that deep structural unity, not computational verification, settles questions about integers.
Is it worth your time
Yes — if you care how deep mathematical proof works, not just that it exists. This is one of the rare cases where a major public claim rested on a single, fragile technical step — and where its repair required not new axioms but a surgical recombination of existing tools. It reveals how modern number theory operates: not through brute calculation, but through layered conditional logic across domains.