# Fermat, Pierre de

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[fairmah] (1601–65) French mathematician: ‘the prince of amateurs’.

As a senior Government law officer it is remarkable that Fermat found time to maintain his skills as a linguist, amateur poet and, most notably, as an amateur mathematician. After 1652, when he nearly died of plague, he did give most of his time to mathematics, but he still did not publish his work in the usual sense, and his results are known through his letters to friends, notes in book margins and challenges to other mathematicians to find proofs for theorems he had devised.

His successes included work on probability, in which he corresponded with and reached agreement with him on some of its basic ideas; on analytical geometry, where again he achieved parallel results with another talented researcher, , and went further in extending the method from two dimensions to three; and on the maxima and minima of curves and tangents to them, where his work was seen by as a starting point for the calculus. In optics he devised Fermat’s principle and used it to deduce the laws of reflection and refraction and to show that light passes more slowly through a dense medium. He worked on the theory of equations and especially on the theory of numbers. Here he was highly inventive and some of his results are well known but, as he usually did not give proofs, they teased other mathematicians in seeking proofs for a long time, with much advantage to the subject. Proofs were eventually found, recently in one case. Fermat’s last theorem, noted in one of his library books, states that the equation *x* *n* + *y* *n* = *z* *n* where *n* is an integer greater than 2, can have no solutions for *x* , *y* and *z* , and records ‘I have discovered a truly marvellous demonstration… which this margin is too narrow to contain’. For over three and a half centuries after he wrote this in about 1637, generations of mathematicians failed to re-create his proof, and some thought that it might be inherently unprovable. However, in 1993 the British mathematician announced a proof. His work has had a great impact on mathematics, as the Last Theorem was also demonstrated to be linked with elliptic equations and modular forms, two topics of wide application in modern mathematics. Fermat’s principle–the dotted line shows the shortest path between A and B. A light beam follows the solid line, consistent with the laws of refraction, because the velocity of light in the glass is less than in air. The solid line is the path of least time.

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