--- In

primenumbers@yahoogroups.com,

"David Broadhurst" <d.broadhurst@...> wrote:

> "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.

The first two had already been noted in

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