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 $p$, $k \ge 1$

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

Basic #

$8=2^3$

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

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

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

A prime is its own first power

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

Two distinct prime factors, so not a prime power

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

Scope #

Listable

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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

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

$9=3^2$

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

Threads over a list; 1 is not a prime power

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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

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

Properties #

$625=5^4$

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

A prime counts as a prime power with exponent 1

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

A prime power need not be squarefree

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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

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

$25=5^2$

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

$16807=7^5$

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

A prime square is composite with Möbius value 0

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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 $-1$

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

$2401=7^4$

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

$121=11^2$

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

Every prime power is deficient

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In$\mathrm{Map}(()↦\mathrm{DivisorSigma}(1, \operatorname{\_1})\lt2\operatorname{\_1}, \mathrm{Filter}(1..100, \mathrm{IsPrimePower}))$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

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

About

Examples	21
Checked against	Wolfram Language 21 agree
Crosswalk	Wolfram Language

See also: IsPrime, FactorInteger