1 reply to this topic

[chrome98]

Giải phương trình nghiệm nguyên:
\[ \phi(\sigma(2^x))=2^x \]

Albania BMO 2009

Edited by chrome98, 07-03-2013 - 07:14.

[kerry0111]

Giải phương trình nghiệm nguyên:
\[ \phi(\sigma(2^x))=2^x \]

Albania BMO 2009

$\phi \left ( \sigma \left ( 2^x \right ) \right )=2^x$

$\Leftrightarrow \phi \left ( 2^n-1 \right )=2^{n-1}$ với $n=x+1$

$\Rightarrow 2^n-1|2^{2^{n-1}}-1$

$\Rightarrow n|2^{n-1}$

$\Rightarrow n=2^m$

$\Rightarrow \phi (2^{2^m}-1)=2^{2^m-1}$

$\Rightarrow 2^{2^m-1}=\phi \left ( F_{m}-2 \right )=\prod_{k=0}^{m-1}\phi \left ( F_{k} \right )\leq \prod_{k=0}^{m-1}2^{2^{k}}=2^{2^m-1}$

$\Rightarrow m-1 \in \left \{ 0;1;2;3;4 \right \}$