Prime power
Encyclopedia
In mathematics
, a prime power is a positive integer power
of a prime number
.
For example: 5=51, 9=32 and 16=24 are prime powers, while
6=2×3, 15=3×5 and 36=62=22×32 are not. The twenty smallest prime powers are:
The prime powers are those positive integers that are divisible by just one prime number; prime powers and related concepts are also called primary
numbers, as in the primary decomposition.
; thus the multiplicative group
of integers modulo pn (or equivalently, the unit group of the ring
) is cyclic
.
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
).
is that the set of prime powers which are not prime is a small set in the sense that the infinite sum
of their reciprocals converges, although the primes are a large set.
(φ) and sigma functions
(σ0) and (σ1) of a prime power are calculated by the formulas:
,
,
.
All prime powers are deficient number
s. A prime power pn is an n-almost prime. It is not known whether a prime power pn can be an amicable number
. If there is such a number, then pn must be greater than 101500 and n must be greater than 1400.
, prime powers play a key role, acting as indicators of lethal dangers in a maze-like cube structure.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a prime power is a positive integer power
Exponentiation
Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...
of a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
.
For example: 5=51, 9=32 and 16=24 are prime powers, while
6=2×3, 15=3×5 and 36=62=22×32 are not. The twenty smallest prime powers are:
- 2, 3, 4, 5, 7, 8, 9, 1111 (number)11 is the natural number following 10 and preceding 12.Eleven is the first number which cannot be counted with a human's eight fingers and two thumbs additively. In English, it is the smallest positive integer requiring three syllables and the largest prime number with a single-morpheme name...
, 1313 (number)13 is the natural number after 12 and before 14. It is the smallest number with eight letters in its name spelled out in English. It is also the first of the teens – the numbers 13 through 19 – the ages of teenagers....
, 1616 (number)16 is the natural number following 15 and preceding 17. 16 is a composite number, and a square number, being 42 = 4 × 4. It is the smallest number with exactly five divisors, its proper divisors being , , and ....
, 1717 (number)17 is the natural number following 16 and preceding 18. It is prime.In spoken English, the numbers 17 and 70 are sometimes confused because they sound similar. When carefully enunciated, they differ in which syllable is stressed: 17 vs 70...
, 1919 (number)19 is the natural number following 18 and preceding 20. It is a prime number.In English speech, the numbers 19 and 90 are often confused. When carefully enunciated, they differ in which syllable is stressed: 19 vs 90...
, 2323 (number)23 is the natural number following 22 and preceding 24.- In mathematics :Twenty-three is the ninth prime number, the smallest odd prime that is not a twin prime. Twenty-three is also the fifth factorial prime, the third Woodall prime...
, 2525 (number)25 is the natural number following 24 and preceding 26.-In mathematics:It is a square number, being 5² = 5 × 5. It is the smallest square that is also a sum of two squares: 25 = 3² + 4²...
, 2727 (number)27 is the natural number following 26 and preceding 28.- In mathematics :Twenty-seven is a perfect cube, being 33 = 3 × 3 × 3. 27 is also 23 . There are exactly 27 straight lines on a smooth cubic surface, which give a basis of the fundamental representation of the E6 Lie algebra...
, 2929 (number)29 is the natural number following 28 and preceding 30.-In mathematics:It is the tenth prime number, and also the fourth primorial prime. It forms a twin prime pair with thirty-one, which is also a primorial prime. Twenty-nine is also the sixth Sophie Germain prime. It is also the sum of three...
, 3131 (number)31 is the natural number following 30 and preceding 32.- In mathematics :Thirty-one is the third Mersenne prime as well as the fourth primorial prime, and together with twenty-nine, another primorial prime, it comprises a twin prime. As a Mersenne prime, 31 is related to the perfect number 496,...
, 3232 (number)32 is the natural number following 31 and preceding 33.-In mathematics:32 is the smallest number n with exactly 7 solutions to the equation φ = n...
, 3737 (number)37 is the natural number following 36 and preceding 38.-In mathematics:It is a prime number, the fifth lucky prime, the first irregular prime, the third unique prime and the third cuban prime of the form...
, 4141 (number)41 is the natural number following 40 and preceding 42.-In mathematics:Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime...
, ...
The prime powers are those positive integers that are divisible by just one prime number; prime powers and related concepts are also called primary
Primary
The word Primary when used alone may refer to any of the following:* Primary , the larger of two co-orbiting bodies* Primary is used for the name of the primary mirror in a telescope.* Primary , from Australia...
numbers, as in the primary decomposition.
Algebraic properties
Every prime power (except powers of 2) has a primitive rootPrimitive root modulo n
In modular arithmetic, a branch of number theory, a primitive root modulo n is any number g with the property that any number coprime to n is congruent to a power of g modulo n. In other words, g is a generator of the multiplicative group of integers modulo n...
; thus the multiplicative group
Multiplicative group of integers modulo n
In modular arithmetic the set of congruence classes relatively prime to the modulus n form a group under multiplication called the multiplicative group of integers modulo n. It is also called the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it...
of integers modulo pn (or equivalently, the unit group of the ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
) is cyclicCyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
.
The number of elements of a finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
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
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
).
Combinatorial properties
A property of prime powers used frequently in analytic number theoryAnalytic 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 Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...
is that the set of prime powers which are not prime is a small set in the sense that the infinite sum
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
of their reciprocals converges, although the primes are a large set.
Divisibility properties
The totient functionEuler's totient function
In number theory, the totient \varphi of a positive integer n is defined to be the number of positive integers less than or equal to n that are coprime to n In number theory, the totient \varphi(n) of a positive integer n is defined to be the number of positive integers less than or equal to n that...
(φ) and sigma functions
Divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships...
(σ0) and (σ1) of a prime power are calculated by the formulas:
,,.All prime powers are deficient number
Deficient number
In number theory, a deficient number or defective number is a number n for which the sum of divisors σIn number theory, a deficient number or defective number is a number n for which the sum of divisors σIn number theory, a deficient number or defective number is a number n for which...
s. A prime power pn is an n-almost prime. It is not known whether a prime power pn can be an amicable number
Amicable number
Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. A pair of amicable numbers constitutes an aliquot sequence of period 2...
. If there is such a number, then pn must be greater than 101500 and n must be greater than 1400.
Popular media
In the 1997 film CubeCube (film)
Cube is a 1997 Canadian science fiction psychological thriller/horror film directed by Vincenzo Natali. The film was a successful product of the Canadian Film Centre's First Feature Project....
, prime powers play a key role, acting as indicators of lethal dangers in a maze-like cube structure.