prime power

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 of integers modulo pⁿ (that is, the group of units of the ring Z/pⁿZ) is cyclic.{{Cite book|title=Prime Numbers: A Computational Perspective

|first1=Richard |last1=Crandall|author1-link=Richard Crandall|first2= Carl B.|last2= Pomerance|author2-link= Carl Pomerance|edition=2nd|publisher=Springer|year=2005|isbn=9780387289793|page=40|url=https://books.google.com/books?id=ZXjHKPS1LEAC&pg=PA40}}

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).{{Cite book|title=A Course in Number Theory and Cryptography|volume=114|series=Graduate Texts in Mathematics|first=Neal|last=Koblitz|author-link=Neal Koblitz|publisher=Springer|year=2012|isbn=9781468403107|page=34|url=https://books.google.com/books?id=4eMlBQAAQBAJ&pg=PA34}}

A property of prime powers used frequently in analytic number theory 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.{{Cite journal |last1=Bayless |first1=Jonathan |last2=Klyve |first2=Dominic |date=November 2013 |title=Reciprocal Sums as a Knowledge Metric: Theory, Computation, and Perfect Numbers |url=https://www.jstor.org/stable/10.4169/amer.math.monthly.120.09.822 |journal=The American Mathematical Monthly |volume=120 |issue=9 |pages=822–831 |doi=10.4169/amer.math.monthly.120.09.822 |jstor=10.4169/amer.math.monthly.120.09.822 |s2cid=12825183 }}

The totient function (φ) and sigma functions (σ₀) and (σ₁) of a prime power are calculated by the formulas

:$\backslash varphi(p^n)\; =\; p^\{n-1\}\; \backslash varphi(p)\; =\; p^\{n-1\}\; (p\; -\; 1)\; =\; p^n\; -\; p^\{n-1\}\; =\; p^n\; \backslash left(1\; -\; \backslash frac\{1\}\{p\}\backslash right),$

:$\backslash sigma\_0(p^n)\; =\; \backslash sum\_\{j=0\}^\{n\}\; p^\{0\backslash cdot\; j\}\; =\; \backslash sum\_\{j=0\}^\{n\}\; 1\; =\; n+1,$

:$\backslash sigma\_1(p^n)\; =\; \backslash sum\_\{j=0\}^\{n\}\; p^\{1\backslash cdot\; j\}\; =\; \backslash sum\_\{j=0\}^\{n\}\; p^\{j\}\; =\; \backslash frac\{p^\{n+1\}\; -\; 1\}\{p\; -\; 1\}.$

All prime powers are deficient numbers. 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.

• Almost prime
• Fermi–Dirac prime
• Perfect power
• Semiprime

{{reflist}}

• Jones, Gareth A. and Jones, J. Mary (1998) Elementary Number Theory Springer-Verlag London {{doi|10.1007/978-1-4471-0613-5}}

{{Classes of natural numbers}}

Category:Prime numbers

Category:Exponentials

Category:Number theory

Category:Integer sequences