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I am studying number theory by my own (I'm not a math major but I'm very interested in this, so I'm sorry I don't know much) and I was wondering if there is a result that says if this next equation has a solution: $$x^a\equiv b \pmod {p^m},$$ with $m, a$ and $b$ given integers, $p$ prime.

I have the feeling this has a solution module $p^m$ if $(a,\phi(p^m))=1$ but I don't know if there is a result for this. Or maybe I'm totally off. If someone has any help, I would appreciate it. The books I'm reading don't have this, or I haven't seen it.

Robert Israel
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user286046
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    To solve this would mean you would be able to extract $a$th roots mod $p^m$, which AFAIK is a hard problem. – Randall Jul 09 '24 at 21:07
  • Okay, thanks, I haven't gotten to that part of the text yet. However, is there for example a particular case for a=3 and p=2? – user286046 Jul 09 '24 at 21:12
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    A solution always exists and is unique if $\gcd(a, \varphi(p^m)) = 1$ and $\gcd(b, p) = 1$, and is given by raising $b$ to the power of $a^{-1} \bmod \varphi(p^m)$. Otherwise solutions may not exist and when they do they may not be unique, but we can say some things about when that happens. – Qiaochu Yuan Jul 09 '24 at 21:23
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    The easy case (cf. prior comment) is described in many answers, e.g. here. $\ \ $ – Bill Dubuque Jul 09 '24 at 21:28
  • @QiaochuYuan Thank you very much. Is there a textbook where I can find this? – user286046 Jul 09 '24 at 21:49
  • @BillDubuque I apologize for not searching more throughly. Thank you for replying, I appreciate it. – user286046 Jul 09 '24 at 21:50

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