Show that $17\mid19^{8n}-1\;\;\forall n\in\mathbb{N}$. I thought about using arithmetic of the remains, proving that $1\equiv19^{8n}\pmod{17}$ And I could not do it: (
2026-04-01 08:03:53.1775030633
$17\mid19^{8n}-1\;\;\forall n\in\mathbb{N}$?
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Hints:
$$19\equiv2\pmod{17}\;,\;\;2^4\equiv-1\pmod{17}\implies$$
$$19^{8n}-1\equiv\left(2^4\right)^{2n}-1\pmod{17}\equiv\ldots$$