Euler Phi Function of 2 times Odd Prime
Jump to navigation
Jump to search
Theorem
Let $n \in \Z_{>0}$ be a semiprime of the form $2 p$, where $p$ is an odd prime.
Let $\map \phi n$ denote the Euler $\phi$ function.
Then:
- $\map \phi n = p - 1$
Proof
By definition $n$ is a semiprime.
As $p$ is an odd prime, $n$ is not square.
Thus from Euler Phi Function of Non-Square Semiprime:
- $\map \phi n = \paren {2 - 1} \paren {p - 1}$
Hence the result.
$\blacksquare$
Examples
$\phi$ of $14$
- $\map \phi {14} = 6$
$\phi$ of $146$
- $\map \phi {146} = 72$
$\phi$ of $362$
- $\map \phi {362} = 180$