Fermat’s Last Theorem is so simple to state, but so hard to prove. Though the 350-year-old claim is a straightforward one about integers, the proof that University of Oxford mathematician Andrew Wiles finally created for it nearly two decades ago required almost unimaginably complex theoretical machinery. The proof was a dazzling demonstration of that machinery’s value, but one aspect of it troubled mathematicians: It relied on stronger axioms than mathematics normally requires, and ones far more complex than are needed to state the problem. Surely, many mathematicians thought, it was possible to prove Fermat’s Last Theorem while assuming less.
Proof — the demonstration of logical consequences arising from a set of axioms — is at the heart of mathematics. But the particular axioms that underlie mathematics aren’t universally agreed upon. The most commonly used axioms are called set theory. But for some theorems, mathematicians assume additional axioms as well. Fewer axioms suffice for others, because set theory involves concepts like infinity that aren’t always needed.
Fermat’s Last Theorem seems too simple to require the full apparatus of set theory, much less even more axioms. Around 1630, Pierre de Fermat noted in the margin of a book that he had discovered a “truly marvelous demonstration” that there are no integers a, b and c that make the equation an + bn = cn true if n is a whole number greater than 2. Unfortunately, he said, the margin was too small for his proof.
When Wiles finally proved the theorem in 1994, he used a deep connection between Fermat’s Last Theorem and algebraic geometry, a field in its infancy in Fermat’s time. Modern algebraic geometry was built using extraordinarily powerful tools developed in the mid-20th century by the mathematician Alexander Grothendieck that rely on an extra axiom in addition to those of standard mathematics — so Wiles’ proof did too.