Natural numbers $n$, $k$ are given such that for any prime $p$ there exists an integer $a$ satisfying the condition $p\mid a^k-n$.

Asked Mar 02 '20 at 11:09

Active Mar 02 '20 at 13:04

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Natural numbers $n$, $k$ are given such that for any prime $p$ there exists an integer $a$ satisfying the condition $p\mid a^k-n$. Decide whether $n$ must necessarily be the $k$-th power of a natural number.

The only thing that occurred to me is that it reminds me of a small Fermat theorem.

edited Mar 02 '20 at 13:04

metamorphy

43,591

asked Mar 02 '20 at 11:09

Tomáš Macháček

1

See also this form where it is assumed that $n$ is $k$'th power mod all $m$ (vs. all prime $m)$. This has a very simple pretty proof – Bill Dubuque Mar 02 '20 at 16:56
@Bill Dobuque but $p \neq x^2$, in your proof – Tomáš Macháček Mar 02 '20 at 17:09
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Yes, the linked question is another form of this question (cf. "this form"). It is helpful to link together closely related questions. – Bill Dubuque Mar 02 '20 at 17:14
@Bill Dobuque I don't understand your proof, could you explained it for me on this problem, please. – Tomáš Macháček Mar 02 '20 at 17:16
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It's best to ask under the proof. Please say precisely which inference is not clear. – Bill Dubuque Mar 02 '20 at 17:21
@Bill Dobuque Could you please write similar answer to this problem like in https://math.stackexchange.com/questions/6976/an-integer-is-a-nth-power-if-that-holds-true-for-all-moduli/1847906#1847906 Please. So I can close it. – Tomáš Macháček Mar 02 '20 at 17:45
@Bill Dobuque So now I understand your proof to https://math.stackexchange.com/questions/6976/an-integer-is-a-nth-power-if-that-holds-true-for-all-moduli/1847906#1847906 but I dont know how transformate or change to my problem. – Tomáš Macháček Mar 02 '20 at 17:52
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It doesn't immediately apply here due to the weaker hypothesis here. – Bill Dubuque Mar 02 '20 at 18:05
So what should I do. – Tomáš Macháček Mar 02 '20 at 18:06

Natural numbers $n$, $k$ are given such that for any prime $p$ there exists an integer $a$ satisfying the condition $p\mid a^k-n$.

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