Diophantine

Diophantine refers to problems and methods associated with Diophantus of Alexandria, a Greek mathematician of the 3rd century. His surviving work, the Arithmetica, explored solving algebraic problems by expressing quantities as ratios of integers and by eliminating surds, laying groundwork for algebraic problem solving. In modern mathematics, the term Diophantine is used to describe equations and problems that seek integer solutions.

A Diophantine equation is a polynomial equation with integer coefficients for which only integer solutions are sought. Simple linear Diophantine equations, such as ax + by = c, can be solved using the greatest common divisor: a solution exists if and only if gcd(a,b) divides c, and all solutions can be described parametrically. Nonlinear examples include Pell’s equation x^2 − Dy^2 = 1, whose solutions are infinite for nonsquare D and can be generated by continued fractions. More general families include exponential Diophantine equations, where unknowns appear as exponents, often with difficult or unresolved solvability questions.

In number theory, Diophantine problems range from concrete equations to broad decidability questions. Fermat’s Last Theorem, asserting that x^n + y^n = z^n has no nontrivial integer solutions for n > 2, is a famous Diophantine statement. A landmark result, Hilbert’s tenth problem, proved that there is no algorithm that decides, for every given Diophantine equation, whether it has an integer solution; this established the limits of algorithmic solvability. The study of Diophantine equations remains active, with connections to algebraic geometry, cryptography, and arithmetic dynamics.