Diophantine equation

In mathematics, a Diophantine equation is an equation in one or more unknowns with integer coefficients for which integer solutions are sought. The word Diophantine refers to the Greek mathematician of the third century A.D., Diophantus of Alexandria, who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra. A linear diophantine equation is one where the unknowns appear only to the first power.

A traditional name for the study of Diophantine equations is Diophantine analysis. The questions asked include:

• Are there any solutions?
• Are there any solutions beyond some that are easily found by inspection?
• Are there finitely or infinitely many solutions?
• Can all solutions be found, in theory?
• Can one in practice compute a full list of solutions?

Such problems often lay unsolved for centuries, and mathematicians gradually came to understand their depth (in some cases), rather than treat them as puzzles. In 1970, a novel result in mathematical logic known as Matiyasevich's theorem showed that it is hopeless to expect a complete theory, in effect settling Hilbert's tenth problem. The point of view of Diophantine geometry, which is the application of algebraic geometry techniques in this field, has continued to grow as a result; since treating arbitrary equations is a dead end, attention turns to equations having a geometric meaning also.

Examples of Diophantine equations are

• ax + by = 1: See Bézout's identity.
• xⁿ + yⁿ = zⁿ: For n = 2 there are 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 Fermat.

One of the few general approaches is through the Hasse principle. Infinite descent is the traditional method, and has been pushed a long way.

The depth of the study of general Diophantine equations is shown by the characterisation of Diophantine sets as recursively enumerable.

The field of Diophantine approximation deals with the cases of Diophantine inequalities: variables are still supposed to be integral, but some coefficients may be irrational numbers, and the equality sign is replaced by upper and lower bounds.