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In mathematics, a Diophantine equation is an indeterminate polynomial equation that only allows the variables to be integers. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. The word Diophantine refers to the Hellenistic mathematician of the 3rd century CE, Diophantus of Alexandria, Egypt who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra. The mathematical study of Diophantine problems Diophantus initiated is now called "Diophantine analysis". A linear Diophantine equation is an equation between two sums of monomials of degree zero or one.

Examples of Diophantine equations


  • ax + by = 1: See Bézout's identity; this is a linear Diophantine.
  • xn + yn = zn: For n = 2 there are infinitely many solutions (x,y,z), the Pythagorean triples. For larger values of n, Fermat's last theorem states that no positive integer solutions x, y, z satisfying the above equation exist.
  • x2 - n y2 = 1: (Pell's equation) which is named, mistakenly, after the English mathematician John Pell. It was studied by Brahmagupta in the 6th century and much later by Fermat.
  • \sum_{i=0}^n{a_i x^i y^{n-i}} = c, where n \geq 3 and c \not= 0: These are the Thue equations, and are, in general, solvable.

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