Re: 3-, 4-, and 5-factor Carmichael families: not terribly common
- --- In firstname.lastname@example.org, "WarrenS" <warren.wds@...> wrote:
>--another remarkable thing about this family by Chernick is that
> (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.
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).