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Question: Show that every integer of the form $(6m + 1)(12m + 1)(18m + 1)$, where m is a positive integer such that $6m + 1$, $12m + 1$, and $18m + 1$ are all primes, is a Carmichael Number

I know that $(6m + 1)(12m + 1)(18m + 1)$ is probably the most widely known form of Carmichael numbers but I'm not entirely sure how to go about proving it for every integer of the form. I was going to write out the proof with the assumption m is prime but going back to reread the question, it seems that m is just a positive integer with no mention of it being prime. Would I factor out the m and write it out from there? Or is there a theorem that I'm missing out on?

CBsmith90
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1 Answers1

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Korselt's criterion says that $n$ is a Carmichael number if and only if $n$ is squarefree, and for every prime $p|n$, we also have $(p-1)|(n-1)$.

For the sample integer, $n=pqr$, where $p=6m+1, q=12m+1, r=18m+1$ are all prime by hypothesis. $n$ is squarefree, being the product of three distinct primes. We calculate and simplify $$n-1=36m(36m^2+11m+1)$$ Hence, each of $p-1, q-1, r-1$ divide $n-1$.

vadim123
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