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 is also a sufficient condition for formula_20 to be prime. Wilson's theorem says that an integer formula_145 is prime if and only if the factorial formula_146 is congruent to formula_147 mod formula_20. For a composite this cannot hold, since one of its factors divides both and formula_149, and so formula_150 is impossible. The formula_20-adic order formula_152 of an integer formula_1 is the number of copies of formula_20 in the prime factorization of formula_1. The same concept can be extended from integers to rational numbers by defining the formula_20-adic order of a fraction formula_157 to be formula_158. The formula_20-adic absolute value