Prime power Prime power In mathematics, a prime power is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The number 1 is not counted as a prime power. The prime powers are: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191,
Prime number 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_195 divides the order of a group, then it has a subgroup of order formula_195. By Lagrange's theorem, any group of prime order is a cyclic group, and by the Burnside theorem any group whose order is divisible by only two primes is solvable. For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications