IsPrimePower

Tests whether $n$ is a positive integer power of a single prime.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguagePrimePowerQ!↗

IsPrimePower(n)$n = p^k$ for a prime $p$, $k \ge 1$

SageMathis_prime_power(n)Juliafirst(is_prime_power_with_data(ZZ(n)))Rustis_prime_power(n)

Typed signatures

Type	Package	Applies
(integer) -> boolean	number-theory	always

Domain: Number theory

Details

• $1=p^0$ does not count: it has no prime base.
• Wolfram's PrimePowerQ.

Examples

4 more kept as data, checked against the oracles

Basic #

$8=2^3$

live ↗

In$\mathrm{IsPrimePower}(8)$Out$\mathrm{True}$

$12=2^2\cdot3$ has two distinct primes

live ↗

In$\mathrm{IsPrimePower}(12)$Out$\mathrm{False}$

A prime is its own first power

live ↗

In$\mathrm{IsPrimePower}(7)$Out$\mathrm{True}$

Two distinct prime factors, so not a prime power

live ↗

In$\mathrm{IsPrimePower}(6)$Out$\mathrm{False}$

Scope #

Listable

live ↗

In$\mathrm{IsPrimePower}(\bigl\lbrack2, 4, 6, 9, 10, 16\bigr\rbrack)$Out$\bigl\lbrack\mathrm{True}, \mathrm{True}, \mathrm{False}, \mathrm{True}, \mathrm{False}, \mathrm{True}\bigr\rbrack$

Large integers

live ↗

In$\mathrm{IsPrimePower}(3^{40})$Out$\mathrm{True}$

$9=3^2$

live ↗

In$\mathrm{IsPrimePower}(9)$Out$\mathrm{True}$

Threads over a list; 1 is not a prime power

live ↗

In$\mathrm{IsPrimePower}(\bigl\lbrack1, 2, 3, 4, 5, 6, 7\bigr\rbrack)$Out$\bigl\lbrack\mathrm{False}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{False}, \mathrm{True}\bigr\rbrack$

Applications #

Prime powers that are not primes

live ↗

In$\mathrm{Filter}(1..100, \operatorname{\_1}↦\mathrm{IsPrimePower}(\operatorname{\_1})∧¬\mathrm{IsPrime}(\operatorname{\_1}))$Out$\bigl\lbrack4, 8, 9, 16, 25, 27, 32, 49, 64, 81\bigr\rbrack$

Properties #

$625=5^4$

live ↗

In$\mathrm{IsPrimePower}(625)$Out$\mathrm{True}$

A prime counts as a prime power with exponent 1

live ↗

In$\mathrm{IsPrimePower}(53)$Out$\mathrm{True}$

A prime power need not be squarefree

live ↗

In$\bigl\lbrack\mathrm{IsPrimePower}(27), \mathrm{IsSquareFree}(27)\bigr\rbrack$Out$\bigl\lbrack\mathrm{True}, \mathrm{False}\bigr\rbrack$

A prime is both a prime power and squarefree

live ↗

In$\bigl\lbrack\mathrm{IsPrimePower}(3), \mathrm{IsSquareFree}(3)\bigr\rbrack$Out$\bigl\lbrack\mathrm{True}, \mathrm{True}\bigr\rbrack$

$25=5^2$

live ↗

In$\mathrm{IsPrimePower}(25)$Out$\mathrm{True}$

$16807=7^5$

live ↗

In$\mathrm{IsPrimePower}(16\,807)$Out$\mathrm{True}$

A prime square is composite with Möbius value 0

live ↗

In$\bigl\lbrack\mathrm{IsPrimePower}(121), \mathrm{IsComposite}(121), \mathrm{MoebiusMu}(121)\bigr\rbrack$Out$\bigl\lbrack\mathrm{True}, \mathrm{True}, 0\bigr\rbrack$

A prime has Möbius value $-1$

live ↗

In$\bigl\lbrack\mathrm{IsPrimePower}(11), \mathrm{IsPrime}(11), \mathrm{MoebiusMu}(11)\bigr\rbrack$Out$\bigl\lbrack\mathrm{True}, \mathrm{True}, -1\bigr\rbrack$

$2401=7^4$

live ↗

In$\mathrm{IsPrimePower}(2\,401)$Out$\mathrm{True}$

$121=11^2$

live ↗

In$\mathrm{IsPrimePower}(121)$Out$\mathrm{True}$

Every prime power is deficient

live ↗

In$\mathrm{Map}(\operatorname{\_1}↦\mathrm{DivisorSigma}(1, \operatorname{\_1})\lt2\operatorname{\_1}, \mathrm{Filter}(1..100, \mathrm{IsPrimePower}))$Out$\bigl\lbrack\mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}\bigr\rbrack$

Possible issues #

1 is $p^0$, which does not count

live ↗

In$\mathrm{IsPrimePower}(1)$Out$\mathrm{False}$

About

Examples	25
Checked against	Wolfram Language 21 agree4 differSageMath 19 agreeSymPy 20 agree3 differJulia 23 agreeRust 17 agree4 errormathlib4 1 errorOscar 1 agree
Crosswalk	Wolfram LanguageSageMathJuliaRust

Implementation

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

See also: IsPrime, FactorInteger