Consider $x \in \Bbb{Z}_p$. Then I want to find the valuation of $(1+p)^x-1$. I think that $val_p((1+p)^x-1)=1+val_p(x)$. Is this right?
Actually I want to prove that
$min\{val_p(1+p)^x-1, val_p(1+p)^{-x}-1\}=1+val_p(x)$
2026-04-01 08:53:57.1775033637
valuation of a particular element in $\Bbb{Z}_p$
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If $p$ is greater than $2$ then your guess is true. Try to find a bound on the valuation of $n!$ and use the binomial expansion.