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Diophantine equations
Main article: Diophantine equation
Fermat's equation xn + yn = zn is an example of a Diophantine equation.[6] A Diophantine equation is a polynomial equation in which thesolutions must be integers.[7] Their name derives from the 3rd-century Alexandrian mathematician, Diophantus, who developed methods for their solution. A typical Diophantine problem is to find twointegers x and y such that their sum, and the sum of their squares, equal two given numbers A and B, respectively:


Diophantus's major work is the Arithmetica, of which only a portion has survived.[8]Fermat's conjecture of his Last Theorem was inspired while reading a new edition of the Arithmetica,[9] which was translated into Latin and published in 1621 by Claude Bachet.[10]
Diophantine equationshave been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation x2 + y2 = z2 are given by the Pythagorean triples, originally solved by the Babylonians (c.1800 BC).[11] Solutions to linear Diophantine equations, such as 26x + 65y = 13, may be found using the Euclidean algorithm (c. 5th century BC).[12] Many Diophantine equations have a form similar tothe equation of Fermat's Last Theorem from the point of view of algebra, in that they have no cross terms mixing two letters, without sharing its particular properties. For example, it is known thatthere are infinitely many positive integers x, y, and z such that xn + yn = zm where n and m are relatively prime natural numbers.[note 1]
[edit]Fermat's conjecture



Problem II.8 in the 1621edition of the Arithmetica of Diophantus. On the right is the famous margin which was too small to contain Fermat's alleged proof of his "last theorem".
Problem II.8 of the Arithmetica asks how a givensquare number is split into two other squares; in other words, for a given rational number k, find rational numbers u and v such that k2 = u2 + v2. Diophantus shows how to solve this sum-of-squares...