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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a prime power is a
positive integer In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
power Power typically refers to: * Power (physics) In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal to one joule per second. In older works, p ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

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 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) has a primitive root; thus the
multiplicative group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of integers modulo ''p''''n'' (i.e. the
group of units In the branch of abstract algebra known as ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations def ...
of the ring Z/''p''''n''Z) is
cyclic Cycle or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in social scienc ...

cyclic
. The number of elements of a
finite field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

isomorphism
).


Combinatorial properties

A property of prime powers used frequently in
analytic number theory 300px, Riemann zeta function ''ζ''(''s'') in the complex plane. The color of a point ''s'' encodes the value of ''ζ''(''s''): colors close to black denote values close to zero, while hue encodes the value's Argument (complex analysis)">argument ...
is that the set of prime powers which are not prime is a small set in the sense that the
infinite sum In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of their reciprocals converges, although the primes are a large set.


Divisibility properties

The
totient function The first thousand values of . The points on the top line represent when is a prime number, which is In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written ...
(''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 number ''n'' for which the sum of divisors of ‘’n’’ is less than 2''n''. Equivalently, it is a number for which the sum of proper divisors (or aliquot sum) is less than ''n' ...
s. A prime power ''pn'' is an ''n''-
almost prime In number theory, a natural number is called ''k''-almost prime if it has ''k'' prime factors. More formally, a number ''n'' is ''k''-almost prime if and only if arithmetic function#Ω(n), ω(n), νp(n) – prime power decomposition, Ω(''n'') = '' ...
. It is not known whether a prime power ''pn'' can be an
amicable number Amicable numbers are two different number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be repre ...
. If there is such a number, then ''pn'' must be greater than 101500 and ''n'' must be greater than 1400.


See also

*
Almost prime In number theory, a natural number is called ''k''-almost prime if it has ''k'' prime factors. More formally, a number ''n'' is ''k''-almost prime if and only if arithmetic function#Ω(n), ω(n), νp(n) – prime power decomposition, Ω(''n'') = '' ...
*
Perfect power In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
*
Semiprime In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...


References

*''Elementary Number Theory''. Jones, Gareth A. and Jones, J. Mary. Springer-Verlag London Limited. 1998. {{Classes of natural numbers Prime numbers Exponentials