Euler's Totient Function Calculator
Counts the number of positive integers up to n that are coprime to n.
Formula first
Overview
Euler's Totient Function, denoted as φ(n), counts the number of positive integers up to n that are relatively prime to n. It is a fundamental multiplicative function in number theory used to explore the properties of modular arithmetic and cyclic groups.
Symbols
Variables
\phi(n) = Totient Value, n = Input Integer
Apply it well
When To Use
When to use: Use this function when calculating the order of the multiplicative group of integers modulo n. It is the primary tool for applying Euler's Theorem in modular exponentiation or when determining the number of generators in a cyclic group of order n.
Why it matters: This equation is the mathematical cornerstone of the RSA encryption algorithm, which secures modern digital communications. It allows for the calculation of private keys by determining the totient of the product of two large primes.
Avoid these traps
Common Mistakes
- Incorrectly including all divisors instead of only unique prime factors in the product formula.
- Confusing phi(n) with the number of divisors au(n).
One free problem
Practice Problem
An analyst needs to determine the number of integers less than 12 that share no common factors with 12 other than 1. Calculate the result of the totient function for this value.
Solve for:
Hint: The prime factors of 12 are 2 and 3.
The full worked solution stays in the interactive walkthrough.
References
Sources
- Wikipedia: Euler's totient function
- Rosen, Kenneth H. Elementary Number Theory and Its Applications. 6th ed. Pearson, 2011.
- A Friendly Introduction to Number Theory by Joseph H. Silverman
- Elementary Number Theory and Its Applications by Kenneth H. Rosen
- Rosen, K. H. (2011). Elementary Number Theory and Its Applications (6th ed.). Pearson.