Find all positive integers $a,b > 1$ that satisfy: $b^a\mid a^b-1$

Asked Dec 09 '24 at 17:12

Active Dec 09 '24 at 17:31

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Find all positive integers $a,b > 1$ that satisfy: $$b^a\mid a^b-1.$$

If I let $p$ be the biggest prime divisor of $b$, I get $$ord_p(a)\mid b$$ $$ord_p(a)\mid\space p-1.$$ Then $ord_p(a)=1$ so $p\mid a-1$, and I got stuck.

edited Dec 09 '24 at 17:31

Anne Bauval

49,005

asked Dec 09 '24 at 17:12

Gianam

Yes, use a\mid b for $a\mid b.$ – Thomas Andrews Dec 09 '24 at 17:15
Please do not use titles consisting only of math expressions; these are discouraged for technical reasons -- see meta. – Anne Bauval Dec 09 '24 at 17:21
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For equality $a^b-1=b^a$ we obtain $(a,b)=(3,2),(2,1)$, see this post. – Dietrich Burde Dec 09 '24 at 17:21
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Using an Approach0 search, in addition to Dietrich's post for the equality case, there's also Solving a Diophantine Equation with 2 variables, which I've just now closed as a duplicate. For your specific question, there are also the AoPS threads Find all integer and Maybe hard number theory (which gives a source of "VMF", and being in the "High School Olympiads" section). – John Omielan Dec 09 '24 at 17:27

Find all positive integers $a,b > 1$ that satisfy: $b^a\mid a^b-1$

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