Show that $t$ is a square $\pmod {2^n} \iff t\equiv 1\pmod{8}$, given that $t$ is odd and $n \ge 3$.
I've tried proving forwards using Hensel's lemma, but got stuck.
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One approach is to show that $x\mapsto x^2$ is $4$-to-$1$ on the units of $\mathbb{Z}/2^n\mathbb{Z}$. – Andrew Dudzik Feb 21 '15 at 18:48
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We haven't covered the theory to do that @Slade – The Problem Feb 21 '15 at 18:51
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Closely related. – Jyrki Lahtonen Feb 21 '15 at 18:53
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And here is how you do it using Hensel's lemma. – Jyrki Lahtonen Feb 21 '15 at 18:55
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1@TheProblem Who needs theory? $x^2 - y^2 = (x-y)(x+y)$ can be divisible by $2^n$ only under a few specific conditions that can be concluded using basic properties of divisibility. – Andrew Dudzik Feb 22 '15 at 04:47