Prove there is none

[Phlembac Adib Hasan]

Prove that there is no positive integer $a$ so that \[13^a\equiv 15\pmod {16}\]

[nayel]

Sorry couldn't resist myself!

$13^1\equiv -3, 13^2\equiv 9, 13^3\equiv -27\equiv 5,13^4\equiv -15\equiv 1,13^5\equiv 13\equiv -3,\dots$ (all mod 16) so $15$ never occurs.

[photon]

$13^a = (3.4+1)^a = (3.4)^a +\binom{a}{1}(3.4)^{a-1} + ......... + \binom{a}{a-1}(3.4) + 1 $ $\equiv \binom{a}{a-1} (3.4) + 1 \not\equiv -1(mod 4^2)$
$\therefore 13^a \not\equiv 15 (mod 4^2)$