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Let $m,n$ be two positive integers and $2^{4n+2}+1\, | \, 2^{4m+2}+1$. Suppose $P_0$ be the largest prime number such that $P_0 \, | \, 2^{4m+2}+1$. If $P_0 \, | \, 2^{4n+2}+1$ then is the following equation true? $$2^{4n+2}+1 = 2^{4m+2}+1$$

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    $\begingroup$ I suspect the answer is "yes"; even for the pair $(2^n+1,2^m+1)$ provided $m>3$. $\endgroup$
    – T. Amdeberhan
    Commented Jan 5, 2017 at 19:16
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    $\begingroup$ @T.Amdeberhan The largest prime factor of $2^{51}+1$ is $43691$, which divides $2^{17}+1$. $\endgroup$
    – Robert Israel
    Commented Jan 5, 2017 at 20:19
  • $\begingroup$ @RobertIsrael: Oh, you're right thanks. I was meaning to write $(2^{2n}+1,2^{2m}+1)$. I Hope this is correct. $\endgroup$
    – T. Amdeberhan
    Commented Jan 5, 2017 at 21:09
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    $\begingroup$ @T. Amdeberhan: That's not right either -- $(n,m)=(58,174)$. $\endgroup$
    – Kevin Buzzard
    Commented Jan 5, 2017 at 21:32

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