The Wolfram Language contains hundreds of original algorithms for computing integer functions involving integers of any size.

Division-Related Functions

Mod  ▪  PowerMod  ▪  Quotient  ▪  QuotientRemainder  ▪  GCD  ▪  LCM  ▪  Divisible  ▪  CoprimeQ  ▪  EvenQ  ▪  OddQ  ▪  PrimeQ  ▪  CompositeQ  ▪  ModularInverse

Combinatorial Functions »

Factorial (!)  ▪  Binomial  ▪  Multinomial  ▪  CatalanNumber  ▪  StirlingS1  ▪  PartitionsP  ▪  BellB  ▪  FrobeniusNumber  ▪  PolygonalNumber  ▪  ...

Recurrence and Sum Functions »

Fibonacci  ▪  LucasL  ▪  HarmonicNumber  ▪  BernoulliB  ▪  EulerE  ▪  MatrixExp  ▪  ...

Integer Decompositions

FactorInteger  ▪  Divisors  ▪  IntegerPartitions  ▪  IntegerDigits  ▪  FrobeniusSolve  ▪  NumberDecompose  ▪  FareySequence

Digit-Related Functions »

IntegerDigits  ▪  FromDigits  ▪  IntegerLength  ▪  NumberDigit  ▪  DigitCount  ▪  IntegerExponent  ▪  IntegerReverse  ▪  ...

Bitwise Functions »

BitAnd  ▪  BitOr  ▪  BitXor  ▪  BitLength  ▪  BitShiftLeft  ▪  BitSet  ▪  ...

Digit-Based Substitution System Functions

ThueMorse  ▪  RudinShapiro

RSolve — solve integer recurrences

Sum — find sums of integer functions

FindSequenceFunction — find functional forms from sequences