In number theory, an algebraic equation in one or more unknowns, with integer coefficients, for which integer solutions are sought. As examples:

1. (i) 14x + 9y = 1 has solutions x = 2 + 9t, y = −3−14t, where t is any integer.

2. (ii) x² + 1 = 2y⁴ has two solutions x = 1, y = 1 and x = 239, y = 13.

3. (iii) x³ + y³ = z³ has only trivial solutions.

See also Catalan’s conjecture, Fermat’s last theorem, Hilbert’s tenth problem, Pell’s equation.