Fermat's last theorem

In number theory, Fermat's Last Theorem, states that no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than two.

For example : when n=2, we can find solutions, a^2 + b^2 = c^2 (Pythagorus)

Example : 3² + 4² = 5² = 25

When n=3, we don’t have any solution

Example : 3^3 + 4^3 = 91 but 3√91 = 4.498

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.

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 was solved by Andrew Wiles in 1995.

Andrew Wiles

Sir Andrew John Wiles, (born 11 April 1953) is a British mathematician and a Royal Society Research Professor at Oxford University, specializing in number theory. He is most famous for proving Fermat's Last Theorem.

Starting in the summer of 1986, Wiles realised that a proof of a limited form of the modularity theorem might then be in reach. He dedicated all of his research time to this problem in relative secrecy. In 1993, he presented his proof to the public for the first time at a conference in Cambridge. In August 1993 it turned out that the proof contained a gap. Wiles tried to fill in this gap, but found out that the error he had made was significant. According to Wiles, the crucial idea for circumventing, rather than closing this gap, came to him on 19 September 1994. Together with his former student Richard Taylor, he published a second paper which circumvented the gap and thus completed the proof. Both papers were published in 1995 in a special volume of the Annals of Mathematics.

Pierre de Fermat

Pierre de Fermat (17 August 1601 or 1607/8 – 12 January 1665) was a French lawyer at the Parlement of Toulouse, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of the differential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for Fermat's Last Theorem, which he described in a note at the margin of a copy of Diophantus'Arithmetica.

Fermat's pioneering work in analytic geometry was circulated in manuscript form in 1636, predating the publication of Descartes' famous La géométrie. This manuscript was published posthumously in 1679 in "Varia opera mathematica", as Ad Locos Planos et Solidos Isagoge, ("Introduction to Plane and Solid Loci").

Fermat was the first person known to have evaluated the integral of general power functions. Using an ingenious trick, he was able to reduce this evaluation to the sum of geometric series. The resulting formula was helpful to Newton, and then Leibniz, when they independently developed the fundamental theorem of calculus.