LaTeX
\varphi(p^k)=p^k-p^{k-1}Variables
- p: prime
- k: positive integer
How to use this formula
Counts positive integers up to a prime power that are coprime to it.
Important notes
- For general n, use multiplicativity across coprime factors.
Quick example
φ(8)=4.
Applicability, worked calculation, and verification
Assumptions and domain checks
- For general n, use multiplicativity across coprime factors.
- Apply the formula over the intended integer domain and check divisibility, coprimality, parity, or modulus conditions.
Worked example
φ(8)=4.
Common mistakes
- Before substituting values into Euler Totient Formula for Prime Powers, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.
- Verify the result of Euler Totient Formula for Prime Powers with a known case, inverse operation, dimensional check, or independent calculation before publishing it.
Continue the workflow
Use Euler Totient Formula for Prime Powers in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
\varphi(p^k)=p^k-p^{k-1}. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Digital Library of Mathematical FunctionsNational Institute of Standards and Technology — Definitions, notation, identities, and reference material for mathematical functions.
Frequently asked questions
What is the Euler Totient Formula for Prime Powers used for?
Counts positive integers up to a prime power that are coprime to it.
Can I copy this formula as LaTeX?
Yes. Copy \varphi(p^k)=p^k-p^{k-1} or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.