Neka su
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
i
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
prirodni brojevi,
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
neparan prost broj, takav da
![p^m \mid a - 1](/media/m/f/1/9/f19d6bf2f76bead35a982113a10a35ec.png)
i
![p^{m+1} \nmid a - 1](/media/m/a/a/7/aa71ceed2438a95a9b62d26689e58b65.png)
. Dokažite da
![p^{m+n} \mid a^{p^n} - 1](/media/m/6/d/8/6d8666426d7f11d97df47c1e1fd7d231.png)
za svaki
![n \in \mathbb{N}](/media/m/2/b/a/2ba27c6141ca415bb86bae1d237f1fac.png)
,
![p^{m+n+1} \nmid a^{p^n} - 1](/media/m/0/d/f/0df7dbb99be4cbc3c3ed621264728a56.png)
za svaki
![n \in \mathbb{N}](/media/m/2/b/a/2ba27c6141ca415bb86bae1d237f1fac.png)
.
%V0
Neka su $a$ i $m$ prirodni brojevi, $p$ neparan prost broj, takav da $p^m \mid a - 1$ i $p^{m+1} \nmid a - 1$. Dokažite da
$a)$ $p^{m+n} \mid a^{p^n} - 1$ za svaki $n \in \mathbb{N}$,
$b)$ $p^{m+n+1} \nmid a^{p^n} - 1$ za svaki $n \in \mathbb{N}$.