Diophantine equation

A polynomial equation whose solutions are sought in integers or another specified number system. Its central challenge is determining which permitted values satisfy the equation, rather than merely finding solutions over the real numbers.

Divisibility: The relation in which one integer is an exact multiple of another. Divisibility tests often rule out candidate solutions or constrain their possible values.
Euclidean algorithm: A repeated-division procedure for computing the greatest common divisor of two integers. It solves linear integer equations and underlies many divisibility arguments.
Linear Diophantine equation: An equation in integers whose variables appear only to the first power. The greatest common divisor gives a complete solvability criterion for a single linear equation.
Diophantus: An ancient Greek mathematician whose surviving work studied equations and numerical solutions. His name became attached to equations whose solutions are restricted to integers or rationals.
Greatest common divisor: The largest positive integer that divides each of a given set of integers. Common divisors determine when linear equations in integers can be solved.
Modular arithmetic: Arithmetic in which integers are identified by their remainders modulo a fixed positive integer. Modular reductions provide quick necessary conditions for integer solutions.
Pythagorean triple: A triple of positive integers satisfying a² + b² = c². This classical equation has a complete parametrization of its integer solutions.
Arithmetica: A Greek mathematical work by Diophantus containing problems about equations and numerical solutions. Its problems helped establish the tradition of seeking numerical solutions to equations.
Congruence (number theory): An equivalence relation stating that two integers have the same remainder modulo a positive integer. Reducing an equation modulo an integer can expose impossible solution patterns.
Infinite descent: A proof method that derives an impossible infinite sequence of decreasing positive integers. It can prove that an equation has no nonzero integer solutions.

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