Euler's phi function

Euler's phi function determines the number of integers less than a given positive integer that are relatively prime to that integer.

Formulas

Given the prime factorization of ${n} = {p}_1^{e_1}{p}_2^{e_2} \cdots {p}_n^{e_n}$ , then one formula for $\phi(n)$ is $\phi(n) = n\left(1-\frac{1}{p_1}\right)\left(1-\frac{1}{p_2}\right) \cdots \left(1-\frac{1}{p_n}\right)$ .

Identities

For prime p, $\phi(p)=p-1$ , because all numbers less than ${p}$ are relatively prime to it.

For relatively prime ${a}, {b}$ , $\phi{(a)}\phi{(b)} = \phi{(ab)}$ .

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Euler's phi function

Formulas

Identities

See also