A225226 - OEIS

%I #22 May 25 2013 05:44:27

%S 3,5,7,13,1093,3511

%N Odd primes p or q with the properties that max(p,q) is Wieferich prime 1093 or 3511 and that binomial(p*q-1,(p*q-1)/2) == binomial(p-1,(p-1)/2)*binomial(q-1,(q-1)/2) (mod (p*q)^2).

%C Same as primes in the pairs {p,q} = {1093,3}, {1093,7}, {1093,13}, {3511,3}, {3511,5}, {3511,13}, and {1093,3511}.

%C If any other primes {p,q} satisfy the congruence, then max(p,q) is a Wieferich prime > 1.2 × 10^13. In that case, min(p,q) might be < 3511.

%C binomial(p*q-1,(p*q-1)/2) == binomial(p-1,(p-1)/2)*binomial(q-1,(q-1)/2) (mod p*q) is a weaker congruence that holds for all odd primes p != q.

%C These results are due to Cai and Granville (2002), as explained in Metsänkylä's review.

%D Tianxin Cai and Andrew Granville, On the residues of binomial coefficients and their products modulo prime powers, Acta Math. Sin., Engl. Ser. 18, No.2 (2002), 277-288.

%H Tauno Metsänkylä, <a href="http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:01885827&amp;type=pdf&amp;format=complete">Review of Cai and Granville's "On the residues of binomial coefficients and their products modulo prime powers"</a>, Zentralblatt 1026.11005.

%Y Cf. Wieferich prime A001220.

%K nonn,fini,full

%O 1,1

%A _Jonathan Sondow_, May 06 2013