Prime power
In
The sequence of prime powers begins:
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, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 251, …
(sequence A246655 in the OEIS).
The prime powers are those positive integers that are divisible by exactly one prime number; in particular, the number 1 is not a prime power. Prime powers are also called primary numbers, as in the primary decomposition.
Properties
Algebraic properties
Prime powers are powers of prime numbers. Every prime power (except
The number of elements of a finite field is always a prime power and conversely, every prime power occurs as the number of elements in some finite field (which is unique up to isomorphism).[2]
Combinatorial properties
A property of prime powers used frequently in
Divisibility properties
The totient function (φ) and sigma functions (σ0) and (σ1) of a prime power are calculated by the formulas
All prime powers are deficient numbers. A prime power pn is an n-almost prime. It is not known whether a prime power pn can be a member of an amicable pair. If there is such a number, then pn must be greater than 101500 and n must be greater than 1400.
See also
References
- ISBN 9780387289793.
- ISBN 9781468403107.
- S2CID 12825183– via JSTOR.
Further reading
- Elementary Number Theory. Jones, Gareth A. and Jones, J. Mary. Springer-Verlag London Limited. 1998.