--- In

primenumbers@yahoogroups.com,

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

> > 1080905, 1739089, 1992641, 2110159, found in only a few minutes

> A few minutes later these turned up: 4013569, 4638985

Proposition: For every integer pair (p,k), we have

V(p, (23*k+11)*3103, 4638985) = p mod 4638985

Proof [using the Sun Tzu Suan Ching]:

v(p,q,n,m)=2*polcoeff(lift(Mod((p+x)/Mod(2,m),x^2+4*q-p^2)^n),0);

n=4638985;

t(p,k,m)=lift(v(p,(23*k+11)*3103,n,m)-p);

s(m)=sum(p=1,m,sum(k=1,m,t(p,k,m)));

if(issquarefree(n),fordiv(n,m,if(isprime(m),print1(s(m)" "))));

0 0 0 0 0

Comment: That gives 65 (q,n) pairs, beating Mike's 37 pairs. But

http://tech.groups.yahoo.com/group/primenumbers/message/21953&var=0
asked, on All-Hallows-Even, for more than 70,000 such pairs,

with n not Carmichael.

David Broadhurst, Hallowmas, 2010