--- In

primenumbers@yahoogroups.com,

"djbroadhurst" <d.broadhurst@...> wrote:

> Proposition: For *every* pair of integers (p,k), we have

> V(p, (11*k + 3)*29*43, 11*29*37*43) = p mod 11*29*37*43

Definition: The integer pair (q,n) is a "Lucas super-pseudoprime"

(LSPS) pair if and only if n is composite, n > q > 0, and

V(p,q,n) = p mod n, for every integer p.

With n = 11*29*37*43 = 507529, there are precisely 37 LSPS pairs

(q,n), namely those with q = (11*k + 3)*29*43, for k = 0 to 36.

Mike Oakes remarked that 507529 is not a Carmichael number.

Puzzle: Find an odd square-free non-Carmichael number

such that there exist more than 70,000 LSPS pairs (q,n).

Comment: There is a solution with n < 10^7.

David Broadhurst, All-Hallows-Even, 2010