Extension to Korselt's Criterion

Question

For a Carmichael number $n$, and $p$ a prime that divides $n$, show that $p-1$ divides $\dfrac{n}{p}-1$. I'm not sure on where to start for this proof, so any help would be appreciated!

Do you mean if $p$ is prime and $p|n$? For instance, since $67*133 = 8911$, you have also have $66|132$ — Umberto P., Mar 15 '15 at 17:50

score 0 · Accepted Answer · edited Apr 13 '17 at 12:21

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If $\ n = pk\ $ then $\ n\!-\!1 = (p\!-\!1)k + k\!-\!1\ $ so $\,p\!-\!1\mid n\!-\!1 \iff p\!-\!1\mid k\!-\!1 = \frac{n}p\!-\!1$

Remark $\ $ A complete proof of Korselt's Criterion is in this answer.

edited Apr 13 '17 at 12:21

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answered Mar 15 '15 at 19:11

Bill Dubuque

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Extension to Korselt's Criterion

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