--- In

primenumbers@yahoogroups.com,

"mikeoakes2" <mikeoakes2@...> wrote:

> if(fac>3

> then the run times collapse drastically

Indeed :-) Here my Chinese speed-up is huge.

Please note that I counted *all* the (q,n) pairs with

square-free non-Carmichael n, coprime to 6, and more

than 3 prime divisors. You stopped when you found

just one q, for a given n, not so?

So you may have missed the dramatic existence of

more than 70,000 (q,n) pairs here:

> n=7056721 [7, 1; 47, 1; 89, 1; 241, 1] is ok

We *already* knew that it was "ok", since q=1 comes from

> "Mike Oakes" Chebyshev NMBRTHRY

What I did not know (but very soon found)

is that for n = 7056721 there are

> precisely 75383 integers q

> such that n > q > 0 and V(p,q,n) = p mod n,

> for every integer p, namely those with

> q = 1 mod 47 and kronecker(q,241) > -1

I still haven't fully digested that very interesting kronecker.

Why does 241 care about the kronecker, while 7 and 89 do not?

David