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Conjecture : $1105$ is the only Poulet-number that can be written in the form $2^a+3^b$ with positive integers $a,b$

A Poulet-number is a composite number $N$ satisfying the congruence $2^{N-1}\equiv 1\mod N$

For $1\le a,b\le 1\ 500$ , the only Poulet-number is $1105$ which is even a Carmichael-number.

If $a=0$ or $b=0$ is also allowed , the composite Fermat-numbers are additional Poulet-numbers of this form. Are those the only Poulet-numbers of the form $2^n+1$ ? If yes , how can we prove this ?

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    $\begingroup$ Downvoter : Which context do you expect here ? $\endgroup$
    – Peter
    Commented Sep 22, 2023 at 13:13
  • 2
    $\begingroup$ I "equalized" the downvote by an upvote :) $\endgroup$
    – Dietrich Burde
    Commented Sep 22, 2023 at 13:15
  • $\begingroup$ Is there any known connection between Poulet and Carmichael Numbers? There aren't that many small Carmichael numbers, so this seems a bit of a coincidence. Still...one example and all. $\endgroup$
    – lulu
    Commented Sep 22, 2023 at 13:27
  • $\begingroup$ @lulu A Carmichael-number is always also a Poulet-number but not vice versa. In fact, I do not expect additional Poulet-numbers. I just am curious whether we can prove it here. $\endgroup$
    – Peter
    Commented Sep 22, 2023 at 13:28
  • $\begingroup$ When you wrote the only Poulet number is $1105$, did you mean the only Poulet number that can be written in the form… ? $\endgroup$
    – J. W. Tanner
    Commented Sep 22, 2023 at 13:50

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