Pierre de Fermat

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Pierre de Fermat IPA: [pjɛːʁ dəfɛʁ'ma] (August 17, 1601 – January 12, 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to modern calculus. 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 then unknown differential calculus, as well as his research into the theory of numbers. He also made notable contributions to analytic geometry and probability.

With his gift for number relations (Diophantus) and his ability to find proofs for his theorems, Fermat essentially created the modern theory of numbers. The quality of his work can be gauged by the fact that many of his results were not proved for over a century after his death, and one of them, his Last Theorem, took more than three centuries to prove. It was the convention among mathematicians in his day to challenge each other to prove a result, often not publishing their own proof to retain an advantage in such competitions.

Although he carefully studied, and drew inspiration from Diophantus, Fermat began a different tradition. Diophantus was content to find a single solution to his equations, even if it were an undesired fractional one. Fermat was interested only in integer solutions to his diophantine equations, and he looked for all possible solutions. He also often proved that certain equations had no solution, which usually baffled his contemporaries.

He studied Pell's equation, Fermat, perfect, and amicable numbers. It was while researching perfect numbers that he discovered the little theorem.

He invented the proof technique of infinite descent, and a factorization method which has been named for him.

He also developed the two-square theorem, and the polygonal number theorem, which states that each number is a sum of three triangular numbers, four square numbers, five pentagonal numbers, and so on.

He 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 theorems of calculus.

Although Fermat claimed to have proved all his arithmetic theorems, few records of his proofs have survived. Many mathematicians, including Gauss, doubted several of his claims, especially given the difficulty of some of the problems and the limited mathematical tools available to Fermat. His famous Last Theorem was first discovered by his son in the margin on his father's copy of an edition of Diophantus, and included the statement that the margin was too small to include the proof. He had not bothered to inform even Mersenne of it.

Together with René Descartes, Fermat was one of the two leading mathematicians of the first half of the 17th century. Independently of Descartes, he discovered the fundamental principles of analytic geometry. With Blaise Pascal, he was a founder of the theory of probability.

Fermat was secretive and a recluse. His only contact with the wider mathematical community aside from a brief exchange of letters with Pascal, was Marin Mersenne. However as Mersenne operated a correspondence network of sorts with other European thinkers, Fermat's results became widely distributed.^[1]

Fermat was born at Beaumont-de-Lomagne, 58 kilometers (36 miles) north-west of Toulouse, France. He died at Castres, 79 kilometers (49 miles) east of Toulouse. The oldest, and most prestigious, college in Toulouse is named after him -- the Pierre de Fermat. The late-15th-century mansion where Fermat was born in Beaumont-de-Lomagne is now a museum.