Euler Phi Function of 83,623,935

Example of Euler $\phi$ Function of Square-Free Integer

$\map \phi {83 \, 623 \, 935} = 41 \, 811 \, 968$

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

Proof

From Euler Phi Function of Square-Free Integer:

$\ds \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$

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

We have that:

$83 \, 623 \, 935 = 3 \times 5 \times 17 \times 353 \times 929$

and so is square-free.

Thus:

\(\ds \map \phi {83 \, 623 \, 935}\)

\(=\)

\(\ds \paren {3 - 1} \paren {5 - 1} \paren {17 - 1} \paren {353 - 1} \paren {929 - 1}\)

\(\ds \)

\(=\)

\(\ds 2 \times 4 \times 16 \times 352 \times 928\)

\(\ds \)

\(=\)

\(\ds 2 \times 2^2 \times 2^4 \times \paren {2^5 \times 11} \paren {2^5 \times 29}\)

\(\ds \)

\(=\)

\(\ds 2^{17} \times 11 \times 29\)

\(\ds \)

\(=\)

\(\ds 41 \, 811 \, 968\)

$\blacksquare$