Euler Phi Function of 33,817,088

Example of Use of Euler $\phi$ Function

$\map \phi {33 \, 817 \, 088} = 16 \, 842 \, 752$

where $\phi$ denotes the Euler $\phi$ Function.

$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$

where $p \divides n$ denotes the primes which divide $n$.

We have that:

$33 \, 817 \, 088 = 2^9 \times 257^2$

Thus:

$\ds \map \phi {33 \, 817 \, 088}$

$=$

$\ds 33 \, 817 \, 088 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 {257} }$

$\ds $

$=$

$\ds 2^9 \times 257^2 \times \frac 1 2 \times \frac {256} {257}$

$\ds $

$=$

$\ds 2^8 \times 257 \times \frac 1 2 \times 2^8$

$\ds $

$=$

$\ds 2^{16} \times 257$

$\ds $

$=$

$\ds 16 \, 842 \, 752$

$\blacksquare$