equation involving only sums, products, and powers in which all the constants are integers and the only solutions of interest are integers. For example, 3x+7y=1 or x²−y²=z³, where x, y, and z are integers. Named in honour of the 3rd-century Greek mathematician Diophantus of Alexandria, these equations were first systematically solved by Hindu mathematicians beginning with Āryabhaṭa I (Aryabhata I) (c. 476–550).

Diophantine equations fall into three classes: those with no solutions, those with only finitely many solutions, and those with infinitely many solutions. For example, the equation 6x−9y=29 has no solutions, but the equation 6x − 9y = 30, which upon division by 3 reduces to 2x−3y=10, has infinitely many. For example, x=20, y=10 is a solution, and so is x=20+3t, y=10+2t for every integer t, positive, negative, or zero. This is called a one-parameter family of solutions, with t being the arbitrary parameter.

Congruence methods provide a useful tool in determining the number of solutions to a Diophantine equation. Applied to the simplest Diophantine equation, ax + by = c, where a, b, and c are nonzero integers, these methods show that the equation has either no solutions or infinitely many, according to whether the greatest common divisor (GCD) of a and b divides c: if not, there are no solutions; if it does, there are infinitely many solutions, and they form a one-parameter family of solutions.