Description
Fermat's Last Theorem (Wiles, 1995): For n > 2, the equation xn + yn = zn has no positive integer solutions. Fermat (1637) claimed a proof; Wiles proved it 358 years later.
Key idea: A non-trivial solution would yield a Frey elliptic curve that is not modular. The Taniyama–Shimura conjecture (proved by Wiles and Taylor–Wiles) states that every elliptic curve over ℚ is modular. Contradiction ⇒ no solution.
Source: Wikipedia
Dependency Flowchart (High-Level)
graph TD
DefEC["Def: Elliptic curve\nE: y² = x³ + ax + b"]
DefMod["Def: Modular form\nHolomorphic, transformation law"]
DefModCurve["Def: Modular elliptic curve\nL(E,s) = L(f,s) for some modular f"]
ThmTS["Thm: Taniyama–Shimura\nEvery E/ℚ is modular"]
LemFrey["Lem: Frey curve\nFLT solution ⇒ non-modular E"]
ThmFLT["Thm: Fermat Last Theorem\nxⁿ + yⁿ = zⁿ has no solution for n>2"]
ThmTS --> ThmFLT
LemFrey --> ThmFLT
DefEC --> LemFrey
DefMod --> DefModCurve
DefModCurve --> ThmTS
DefEC --> ThmTS
classDef definition fill:#b197fc,color:#fff,stroke:#9775fa
classDef theorem fill:#51cf66,color:#fff,stroke:#40c057
classDef lemma fill:#74c0fc,color:#fff,stroke:#4dabf7
class DefEC,DefMod,DefModCurve definition
class ThmTS,ThmFLT theorem
class LemFrey lemma