 The question of how many integer prime numbers factor into a product of multiple prime ideals in an algebraic number field is addressed by Chebotarev's density theorem, which (when applied to the cyclotomic integers) has Dirichlet's theorem on primes in arithmetic progressions as a special case.In the theory of finite groups the Sylow theorems imply that, if a power of a prime number formula_198 divides the order of a group, then it has a subgroup of order formula_198.By Lagrange's theorem, any group of prime order is a cyclic group,
and by Burnside's theorem any group whose order is divisible by only two primes is solvable.