Friday, 3 January 2014

Fermat's Last Theorem

In 1995, British mathematician solved the 358-year-old mathematical theory, Fermat's Last Theorem, which was widely regarded as the most difficult maths problem in the world. 

Fermat's Fermat's Last Theorem is a theory written by Pierre de Fermat in 1637 that states that no three positive integers a, b, and c can satisfy the equation an + bn = c for any integer value of n greater than two.

Fermat famously wrote the theory in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. No successful proof was published until 1995 despite the efforts of countless mathematicians during the 358 intervening years. 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. It is among the most famous theorems in the history of mathematics and prior to its 1995 proof by Andrew Wiles it was in the Guinness Book of World Records for "most difficult mathematical problems".

The Theorem itself is a deceptively simple statement within mathematics that Fermat famously stated he had solved around 1637. His claim was discovered some 30 years later, after his death, as a bare statement in the margin of a book, but Fermat died without leaving any proof of his claim.

The claim eventually became one of the most famous unsolved problems of mathematics. The attempts made to prove it during that time prompted substantial development of number theory and over time Fermat's Last Theorem itself gained legendary prominence as an unsolved problem in popular mathematics. It is based upon the well known formula ("Pythagoras' Theorem") for a right-angle triangle discovered by the ancient Greek mathematician Pythagoras: a2 + b2 = c2

The Pythagorean equation has an infinite number of whole-number solutions, representing the sides of a right-angle triangle; these solutions are known as Pythagorean triples. Fermat conjectured that the more general equation an + bn = cn had no solutions in positive integers a, b and c for any integer greater than 2 — in other words that although a2 + b2 = c2 had an infinite number of whole-number solutions, the similar equations
a3 + b3 = c3
a4 + b4 = c4
an + bn = cn
for any other integer exponent n greater than 2 would have no solutions in positive integers. Although he claimed to have a general proof of his conjecture, Fermat left no details of his proof apart from the special case n = 4.

In 1816 and again in 1850, the French Academy of Sciences offered a prize for a general proof of Fermat's Last Theorem. In 1857, the Academy awarded 3000 francs and a gold medal to Kummer for his research on ideal numbers, although he had not submitted an entry for the prize.[127] Another prize was offered in 1883 by the Academy of Brussels.[128]

In 1908, the German industrialist and amateur mathematician Paul Wolfskehl bequeathed 100,000 marks to the Göttingen Academy of Sciences to be offered as a prize for a complete proof of Fermat's Last Theorem. On 27 June 1908, the Academy published nine rules for awarding the prize. Among other things, these rules required that the proof be published in a peer-reviewed journal; the prize would not be awarded until two years after the publication; and that no prize would be given after 13 September 2007, roughly a century after the competition was begun. Wiles collected the Wolfskehl prize money, then worth $50,000, on 27 June 1997.

Prior to Wiles' proof, thousands of incorrect proofs were submitted to the Wolfskehl committee, in the first year alone (1907–1908), 621 attempted proofs were submitted, although by the 1970s, the rate of submission had decreased to roughly 3–4 attempted proofs per month. Mathematical historian Howard Eves even said, "Fermat's Last Theorem has the peculiar distinction of being the mathematical problem for which the greatest number of incorrect proofs have been published."