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## Re: 3-, 4-, and 5-factor Carmichael families: not terribly common

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• ... --another remarkable thing about this family by Chernick is that its change the sign of 1 version N = (6 x - 1) (12 x - 1) (18 x - 1) is another kind of
Message 1 of 6 , Dec 13, 2012
--- In primenumbers@yahoogroups.com, "WarrenS" <warren.wds@...> wrote:
>
> (6 x + 1) (12 x + 1) (18 x + 1)
> is a Carmichael number if all three factors are simultaneously
> prime and where x>0 is integer.

--another remarkable thing about this family by Chernick is that
its "change the sign of 1" version
N = (6 x - 1) (12 x - 1) (18 x - 1)
is another kind of interesting pseudoprime if all three factors are simultaneously prime and where x>0 is integer.

Namely, every prime P dividing N has property that P+1 divides N+1;
this causes certain Lucas-like primality tests to always think N is a prime (unless they find a factor).

----

If N enjoyed both this property and the usual Carmichael property simultaneously, then N would be (in my terminology) a "2-generalized Carmichael."

Broadhurst provided a list by Robert Gerbicz 1 April 2009,
which it turns out was not a joke, of 144153 numbers N.
Also, Broadhurst provided his own list of 246 numbers N.
Unfortunately none of the members of either list
satisfy my criteria for being 2-generalized Carmichael --
that is, I demand that if P is prime and P|N, then (P+1)|(N+1) and (P-1)|(N-1).
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