Diophantine Equations

Integer solutions to polynomial equations — from Pythagorean triples to Fermat's Last Theorem (Wiles, 1995)

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Fermat's Last Theorem: x^n + y^n = z^n has no solutions in ℤ⁺ for n ≥ 3

Proven by Andrew Wiles in 1995 using modularity of elliptic curves. The key insight: a solution would yield a Frey elliptic curve that cannot be modular, contradicting the Taniyama-Shimura-Weil conjecture. Pell's equation x²−Dy²=1 always has infinitely many solutions (fundamental solution via continued fractions).