mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a prime power is a

positive integer In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...

which is a positive integer power of a single

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

. For example: , and are prime powers, while , and are not. 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, …

. The prime powers are those positive integers that are

divisible In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...

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 In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many ''primary ideals'' (which are related ...

Properties

Algebraic properties

Prime powers are powers of prime numbers. Every prime power (except powers of 2 greater than 4) has a primitive root; thus the

multiplicative group In mathematics and group theory, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...

of integers modulo ''p''^''n'' (that is, the

group of units In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that vu = uv = 1, where is the multiplicative identity; the ele ...

of the ring Z/''p''^''n''Z) is cyclic. The number of elements of a

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...

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 In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

Combinatorial properties

A property of prime powers used frequently in

analytic number theory In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dir ...

is that the

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

of prime powers which are not prime is a small set in the sense that the

infinite sum In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathemati ...

of their reciprocals converges, although the primes are a large set.

Divisibility properties

The totient function (''φ'') and sigma functions (''σ''₀) and (''σ''₁) of a prime power are calculated by the formulas :

\varphi(p^n) = p^ \varphi(p) = p^ (p - 1) = p^n - p^ = p^n \left(1 - \frac\right),

\sigma_0(p^n) = \sum_^ p^ = \sum_^ 1 = n+1,

\sigma_1(p^n) = \sum_^ p^ = \sum_^ p^ = \frac.

All prime powers are

deficient number In number theory, a deficient number or defective number is a positive integer for which the sum of divisors of is less than . Equivalently, it is a number for which the sum of proper divisors (or aliquot sum) is less than . For example, th ...

s. 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.

Properties

Algebraic properties

Combinatorial properties

Divisibility properties

See also

References

Further reading