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👉 The landmark 2018 proof of Fermat's Last Theorem (FLT) by Andrew Wiles is a triumph of modern number theory, combining deep algebraic geometry and modular forms. Wiles focused on proving a special case of FLT: for prime exponents
p
where 3 ≤
p
≤ 1000, the equation
a^p + b^p = c^p
has no non-trivial integer solutions. He leveraged his earlier work on elliptic curves and modularity, showing that a hypothetical solution would force a non-modular elliptic curve to arise—a contradiction. By connecting FLT to the Taniyama-Shimura conjecture (now modularity theorem), Wiles demonstrated that non-modular elliptic curves cannot exist, thereby proving FLT for the targeted primes. This required intricate techniques like Galois representations and deformation theory, marking a pivotal synthesis of algebraic geometry and number theory.

winning math