The traditional interest in Pythagorean triples connects with the Pythagorean theorem in its converse form, it states that a triangle with sides of lengths a, b, and c has a rightįermat's Last Theorem angle between the a and b legs when the numbers are a Pythagorean triple. There are infinitely many such triples, and methods for generating such triples have been studied in many cultures, beginning with the Babylonians and later ancient Greek, Chinese, and Indian mathematicians. Mathematical contextPythagorean triplesPythagorean triples are a set of three integers (a, b, c) that satisfy a special case of Fermat's equation (n=2) Examples of Pythagorean triples include (3, 4, 5) and (5, 12, 13). Wiles's achievement was reported widely in the popular press, and has been popularized in books and television programs. Building on work of Ken Ribet, Andrew Wiles succeeded in proving enough of the modularity theorem to prove Fermat's Last Theorem, with the assistance of Richard Taylor. In 1984, Gerhard Frey suggested the approach of proving the conjecture through a proof of the modularity theorem for elliptic curves. The final proof of the conjecture for all n came in the late 20th century. Building on Kummer's work and using sophisticated computer studies, other mathematicians were able to prove the conjecture for all odd primes up to four million.
In the mid-19th century, Ernst Kummer proved the theorem for regular primes. Over the next two centuries (16371839), the conjecture was proven for only the primes 3, 5, and 7, The 1670 edition of Diophantus' Arithmetica includes Fermat's commentary, particularly his "Last Theorem" (Observatio although Sophie Germain proved a special case for all Domini Petri de Fermat). This reduced the problem to proving the theorem for exponents n that are prime numbers. It is among the most famous theorems in the history of mathematics and prior to its 1995 proof was in the Guinness Book of World Records for "most difficult mathematical problems".įermat's conjecture (history)Fermat left no proof of the conjecture for all n, but he did prove the special case n=4. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. No successful proof was published until 1995 despite the efforts of countless mathematicians during the 358 intervening years. This theorem was first conjectured by Pierre de Fermat in 1637, famously in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. Fermat's Last TheoremIn number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an+bn=cn for any integer value of n greater than two.