$\frac{a^n +b^n}{(ab)^{n-1}+1}$ is a perfect $n^{th}$ power

Asked Apr 09 '11 at 12:38

Active Jan 09 '20 at 20:47

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Let $a,b$ be positive integers satisfying $$(ab)^{n-1}+1 \mid a^n +b^n.$$ Then how to show that the number $\frac{a^n +b^n}{(ab)^{n-1}+1}$ is a perfect $n^{th}$ power of an integer?

Another question: is this problem true for $a,b \in \mathbb Z$?

PS. posted to be connected with this.

edited Jan 09 '20 at 20:47

Alejandro Tolcachier

1,938

asked Apr 09 '11 at 12:38

Amir Parvardi

4,896

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What is the source of the problem? – Bill Dubuque Apr 09 '11 at 15:46
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See http://www.artofproblemsolving.com/Forum/viewtopic.php?f=56&t=349211 – Grigory M Apr 09 '11 at 16:17
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@Gri The question was to Amir, in attempt to understand his motivation. Alas, this is completely absent in all his questions. If he already knows these answers to the many well-known competition problems that he posed here, then why is he asking the questions here? – Bill Dubuque Apr 09 '11 at 16:32
@Bill My comment was also meant for Amir, actually – Grigory M Apr 09 '11 at 17:11
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@Gri If you follow his link above you'll see the that Amir already knows about that AoPS page (and many more). Hence my above remark. – Bill Dubuque Apr 09 '11 at 17:13
Yes, I know about the AoPS page, but I liked to see other solutions. What about that "Another question"? – Amir Parvardi Apr 09 '11 at 17:17

$\frac{a^n +b^n}{(ab)^{n-1}+1}$ is a perfect $n^{th}$ power

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