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• ## Re: Carmichael numbers of order 2

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• ... Richard Finch calls these unusually strong Lucas-Carmichael-minus (uLC-) numbers, with p^2-1|N-1, for every prime p|N. See:
Apr 3, 2009 1 of 12
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> "What are the first 3 Carmichael numbers of order 2?"
>
> 443372888629441
> 39671149333495681
> 842526563598720001

Richard Finch calls these
"unusually strong Lucas-Carmichael-minus" (uLC-) numbers,
with p^2-1|N-1, for every prime p|N. See:
http://www.chalcedon.demon.co.uk/rgep/p20.pdf

I found the third of these by mining Richard's file of
Carmichael numbers between 10^17 than 10^18.
http://www.chalcedon.demon.co.uk/rgep/cartable.html

Richard classified
582920080863121 = 41 * 53 * 79 * 103 * 239 * 271 * 509
as a "strong", but not "unusually strong",
Lucas-Carmichael-minus number
since in this case both p-1 and p+1
divide N-1, for each prime p|N, but p^2-1
does not divide N-1 in the cases p = 79, 239, 271,
where p^2 = 1 mod 2^5
whilst N = 17 mod 2^5.

If we ask merely that p-1 and (p+1)/2 divide N-1,
then the following 3 numbers also occur, for N < 10^18:
28295303263921
894221105778001
2013745337604001
making 7 in all, of which only

842526563598720001
= 17 * 61 * 71 * 89 * 197 * 311 * 769 * 2729

occurs for 10^18 > N > 10^17.

Finally, I remark that Richard found precisely one
"unusually strong Lucas-Carmichael-plus" (uLC+) number,
with p^2-1|N+1, for prime p|N and N < 10^13, namely

79397009999 = 23 * 29 * 41 * 43 * 251 * 269.

David
• Oh dear, another silly typo: I meant Richard Pinch, of course.
Apr 3, 2009 1 of 12
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Oh dear, another silly typo: I meant Richard Pinch, of course.
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