How would I find, given an integer $n$, for which values of $m$ (also integer) does $x^n \equiv y^n \pmod{m}$ imply $x \equiv y \pmod{m}$?
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No $m$ satisfies this condition. You probably mean $x\equiv y$ instead of $x=y$. Also, this question is not about abstract algebra. – Kemono Chen Oct 29 '18 at 03:52
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Yes, that's what I meant, sorry. – user609740 Oct 29 '18 at 03:53
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Suppose that $m$ happens to satisfy $\gcd(\phi(m),n)=1$. Does that allow you to prove the desired implication? – Greg Martin Oct 29 '18 at 04:13