Although Diophantine equations provide classic examples of undecidability, Mathematica in practice succeeds in solving a remarkably wide range of such equations—automatically applying dozens of often original methods, many based on the latest advances in number theory.
FindInstance — find particular solutions to Diophantine equations
Reduce — general reduction of Diophantine equations and inequalities
IntegerPartitions — partitions of an integer
FrobeniusSolve — solve Frobenius equations
ChineseRemainder — solve simultaneous linear congruences