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Diophantine Equations
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 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
SquaresR  ▪ PowersRepresentations  ▪ PartitionsP  ▪ ...
ChineseRemainder solve simultaneous linear congruences
MultiplicativeOrder  ▪ PowerMod  ▪ ...
Minimize, Maximize integer optimization