Euler Phi Function Of 666 Equals Product Of Digits - ProofWiki

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Theorem

The number $666$ has the following interesting property:

$\map \phi {666} = 6 \times 6 \times 6$

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

From Euler Phi Function of Integer:

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

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

We have that:

$666 = 2 \times 3^2 \times 37$

Thus:

$\ds \map \phi {666}$	$=$	$\ds 666 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 3} \paren {1 - \dfrac 1 {37} }$
$\ds $	$=$	$\ds 666 \times \frac 1 2 \times \frac 2 3 \times \frac {36} {37}$
$\ds $	$=$	$\ds 3 \times 1 \times 2 \times 36$
$\ds $	$=$	$\ds 3 \times 2 \times \paren {2^2 \times 3^2}$
$\ds $	$=$	$\ds 216$
$\ds $	$=$	$\ds 6 \times 6 \times 6$

$\blacksquare$

1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $666$

Từ khóa » Phi 666

\(\ds \map \phi {666}\)	\(=\)	\(\ds 666 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 3} \paren {1 - \dfrac 1 {37} }\)
\(\ds \)	\(=\)	\(\ds 666 \times \frac 1 2 \times \frac 2 3 \times \frac {36} {37}\)
\(\ds \)	\(=\)	\(\ds 3 \times 1 \times 2 \times 36\)
\(\ds \)	\(=\)	\(\ds 3 \times 2 \times \paren {2^2 \times 3^2}\)
\(\ds \)	\(=\)	\(\ds 216\)
\(\ds \)	\(=\)	\(\ds 6 \times 6 \times 6\)