gives the Euler totient function .
- EulerPhi is also known as the Euler totient function or phi function.
- Integer mathematical function, suitable for both symbolic and numerical manipulation.
- Typically used in cryptography and in many applications in elementary number theory.
- EulerPhi[n] counts positive integers up to n that are relatively prime to n.
- For a number with a unit and primes, EulerPhi[n] gives .
Examplesopen allclose all
Numerical Manipulation (4)
Basic Applications (4)
Number Theory (5)
Plot the cumulative sum of EulerPhi:
The probability that two randomly chosen positive integers less than x are relatively prime:
Properties & Relations (11)
EulerPhi is non-negative:
EulerPhi is a multiplicative function:
Similarly, EulerPhi[n]==n∏p|n(1-1/p) where p is prime:
Determine EulerPhi through prime factorization:
Possible Issues (1)
Value at 0:
Neat Examples (4)
Form an absolutely abnormal number as the limit of the following sequence:
Digits of the sixth approximation in various bases:
Iterate the map and display result modulo :
Plot of Ulam spiral where numbers are colored based on the values of EulerPhi:
Wolfram Research (1988), EulerPhi, Wolfram Language function, https://reference.wolfram.com/language/ref/EulerPhi.html (updated 2007).
Wolfram Language. 1988. "EulerPhi." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007. https://reference.wolfram.com/language/ref/EulerPhi.html.
Wolfram Language. (1988). EulerPhi. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/EulerPhi.html