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I'm trying to understand a proof for a theorem that states conditions under which the group of primitive residual classes modulo $n$ is cyclic. This proof uses the following result attributed to Gauß: $$ (1 + p)^{p^{\nu - 1}} \equiv 1 \pmod {p^\nu},\qquad(1 + p)^{p^{\nu - 1}} \not\equiv 1 \pmod {p^{\nu + 1}}, $$ where $p$ is an odd prime and $\nu \in \mathbb{N}$.

This result is used to derive without further explanation the following incongruence $$ (1 + p)^{p^{\nu - 2}} \not\equiv 1 \pmod {p^{\nu}} $$ where $\nu \geq 2$.

Can somebody explain how one can obtain the second incongruence from the first one? It's probably something simple, but I'm just not seeing it at the moment. Thanks.

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I think the bottom line is just the first with $\nu$ shifted by one.

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  • $\begingroup$ Arrghh, of course. Thanks. $\endgroup$
    – Markus
    Commented Dec 23, 2012 at 19:05

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