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Carmichael numbers are composite $n$ for which $$a^{n-1}\equiv1\quad(mod\ n)$$ is true for every prime $a<n$.

Part of a proof I'm currently working through includes the condition that for Carmichael numbers $n$, $(p_i -1)\mid(n-1)$ is true for every prime factor $p_i$.

I've tested this for some Carmichael numbers and it is true while for non-Carmichael numbers it not always appears to hold. I just don't quite understand why this is true.

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  • Have you tried to prove it? – jjagmath May 30 '24 at 13:28
  • I currently have no idea how to start. But it has to be something really fundamental because it was mentioned without any further note in the proof I saw it – 299792458 May 30 '24 at 13:40
  • From How to ask a good question: "Avoid "no clue" questions. Too many questions begin or end with "I don't even know how to begin with this problem". While this may be true (you may genuinely have no idea how to approach the problem), it is still not a valid reason to limit your post to the statement of the problem without any mention of your own thoughts. Such questions will most of the time be rejected by the community, which represents a significant waste of time - including for yourself ..." – jjagmath May 30 '24 at 13:45

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