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# Fermat and his Last Theorem

"It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain."

Pierre de Fermat (pronounced Fair-mah),  (born August 17, 1601, Beaumont-de-Lomagne, France—died January 12, 1665, Castres), French mathematician who is often called the founder of the modern theory of numbers. Together with René Descartes, Fermat was one of the two leading mathematicians of the first half of the 17th century. Independently of Descartes, Fermat discovered the fundamental principle of analytic geometry. His methods for finding tangents to curves and their maximum and minimum points led him to be regarded as the inventor of the differential calculus. Through his correspondence with Blaise Pascal he was a co-founder of the theory of probability.

He is best known for Fermat's Last Theorem, which he described in a note at the margin of a copy of Diophantus' Arithmetica. In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that the equation an + bn = cn  has no non-zero integer solutions for x, y and z when n > 2.
French lawyer Fermat pursued mathematics in his spare time. Although he pursued mathematics as an amateur, his work in number theory was of such exceptional quality and erudition that he is generally regarded as one of the greatest mathematicians of all times. He had the habit of scribbling notes in the margins of books or in letters to friends rather than publishing them.

Let's consider the equation a2 + b2 = c2 . We know that one can have integer solutions for this equation i.e. 32 + 42 = 52. etc. Fermat also knew this. But suddenly something new striken his mind when he was studying Diophantus' Arithmetica (2nd volume),  If we replace this value of n with an integer  greater than 2, can we still have a integer solution for a, b, c satisfying this equation ?

As a result of Fermat's marginal note, the proposition that the Diophantine equation an + bn = cn  has no non-zero integer solutions for x, y and z when n > 2. has come to be known as Fermat's Last Theorem.

The full text of Fermat's statement, written in Latin, reads

"Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet" (Nagell 1951, p. 252). In translation, "It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain."

These marginal notes only became known after Fermat's son Samuel published an edition of Bachet's translation of Diophantus's Arithmetica with his father's notes in 1670.

Since 1637 the Mathematicians were trying to solve this theorem. But it is unnecessary to mention that their effort were in vain. Though Swiss Mathematician Euler proved it for n= 3 and 4; Legendre for  n=5. But, no one was able to generalized the theorem for all the cases of n.

It is now believed that Fermat's 'proof' was wrong although it is impossible to be completely certain. The truth of Fermat's assertion was proved in June 1993 by the British mathematician Andrew Wiles, but Wiles withdrew the claim to have a proof when problems emerged later in 1993. In November 1994 Wiles again claimed to have a correct proof which has now been accepted.

Unsuccessful attempts to prove the theorem over a 300 year period led to the discovery of commutative ring theory and a wealth of other mathematical discoveries.

Contributions of Fermat:

• Fermat is considered to be one of the 'fathers' of analytic geometry. (Along with Rene' Descartes.)
• Fermat along with Blaise Pascal is also considered to be one of the founders of probability theory.
• Fermat also made contributions in the field of optics and provided a law on light travel and made wrote a few papers about calculus well before Isaac Newton and Gottfried Leibniz were actually born.
• The Most Famous Question In Math History for 350 Years!  Fermat's Last Theorem !!!

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