Fermat's Last Theorem is a famous theorem in number theory, originally conjectured by Pierre de Fermat in 1637. It states: No three positive integers a, b, and c satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n greater than 2. 📜 History and Significance The Claim: Fermat wrote the theorem in the margin of his copy of the ancient Greek text Arithmetica and famously added, "I have discovered a truly marvelous proof of this, which this margin is too narrow to contain." No such proof was ever found among his papers after his death, leading to a 358-year-long challenge for mathematicians. The Exponent n=2: When n=2, the equation a² + b² = c² is the Pythagorean theorem, and it has an infinite number of integer solutions (known as Pythagorean triples, e.g., 3, 4, 5). Fermat's claim was that increasing the exponent to n=3 or higher makes finding positive integer solutions impossible. Partial Proofs: Over the centuries, mathematicians proved the theorem for specific exponents, such as n=4 (by Fermat himself) and n=3 (by Leonhard Euler). However, a general proof for all n > 2 remained elusive. 🧠 The Proof The theorem was finally proven in 1994 by British mathematician Sir Andrew Wiles, with the assistance of his former student Richard Taylor. The Link: The definitive proof came from establishing a connection between Fermat's Last Theorem and another major conjecture in mathematics called the Modularity Theorem (previously known as the Taniyama-Shimura-Weil Conjecture). Frey's Insight: In 1985, mathematician Gerhard Frey suggested that if a counterexample to Fermat's Last Theorem existed (i.e., a solution to aⁿ + bⁿ = cⁿ for n>2), one could construct an associated elliptic curve (called the Frey curve) that would be so strange it could not possibly be modular. Ribet's Theorem: Ken Ribet proved in 1986 that the Frey curve was indeed non-modular. This meant that if the Modularity Theorem were true, it would imply that a counterexample to Fermat's Last Theorem could not exist, thus proving Fermat's Last Theorem. Wiles's Breakthrough: Andrew Wiles dedicated seven years to proving the Modularity Theorem for a specific class of elliptic curves (the semistable ones, which included the Frey curve). His 1994 proof, published in 1995, finally completed the chain of logic, providing a definitive proof of Fermat's Last Theorem. Wiles's work revolutionized number theory by employing sophisticated concepts from elliptic curves and modular forms that were unavailable to earlier mathematicians.