> > http://alumnus.caltech.edu/~however/talks/FortCollins.pdf

--None of the three numbers listed in http://oeis.org/A175531

>

> http://oeis.org/A175531 gives an update

>

> David

qualifies as a "2-generalized Carmichael number" by my definition.

Howe's definition on page 16 of his lecture slides is different from

and I'm not even sure it is related to mine.

I don't understand Howe's motivation, although I guess it must be

spiritually similar to mine.

I suspect a 2-generalized Carmichael exists with N<10^100.- --- In primenumbers@yahoogroups.com,

"WarrenS" <warren.wds@...> wrote:> Howe's definition on page 16 of his lecture slides is different

Yes. His was more interesting to me, in the context

of pseudoprimality:

https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1006&L=nmbrthry&T=0&F=&S=&P=51

http://tech.groups.yahoo.com/group/primenumbers/message/21461

It is easy to construct larger solutions. For example

http://tech.groups.yahoo.com/group/primenumbers/message/19971

provides a link to 246 solutions in

http://physics.open.ac.uk/~dbroadhu/cert/erdos2.out

with n = 1 mod p^2-1, for each prime p|n.

For 144153 such solutions, see the 12 MB file

http://physics.open.ac.uk/~dbroadhu/cert/carmrob2.txt

obtained by Robert Gerbicz on 1 April 2009

David - Ouch, there was a typo in the definition. Here is the message back again

with typo corrected (k-1 changed to k):

It seems the following concept ought to be important:

"k-Generalized Carmichael numbers."

DEFINITION:

If N is composite and squarefree and:

for all primes p that divide N,

p-1 happens to divide N-1,

and:

for each j=2,3,4,...,k,

for all primes p that divide N,

(p^j-1)/(p-1)

happens to divide (N^j-1)/(N-1),

then call N a "k-generalized Carmichael number."

If you make k large enough as a function of N, then I guess

k-generalized Carmichael numbers must stop existing.

QUESTION: how large do you need?

As an initial guess, if k>(logN)*(some positive constant)

that is enough to stop N from being a

k-Generalized Carmichael number.

QUESTION:

Can you find any k-generalized Carmichaels for k=2,3 or 4?

[I believe my computer showed none exist when N<10^16.]