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Diophantine equations

Fermat's equation, x + y = z with positive integer solutions, is an example of a Diophantine equation, named for the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers 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. Fermat's conjecture of his Last Theorem was inspired while reading a new edition of the Arithmetica, that was translated into Latin and published in 1621 by Claude Bachet.

Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation x + y = z are given by the Pythagorean triples, originally solved by the Babylonians (c. 1800 BC). Solutions to linear Diophantine equations, such as 26 x + 65 y = 13, may be found using the Euclidean algorithm (c. 5th century BC). Many Diophantine equations have a form similar to the 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 that there are infinitely many positive integers x, y, and z such that x + y = z, where n and m are relatively prime natural numbers.