A225226 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).

3, 5, 7, 13, 1093, 3511

Offset

1,1

Comments

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

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.

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.

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

References

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.

Links

Table of n, a(n) for n=1..6.

Tauno Metsänkylä, Review of Cai and Granville's "On the residues of binomial coefficients and their products modulo prime powers", Zentralblatt 1026.11005.

Crossrefs

Cf. Wieferich prime A001220.

Sequence in context: A064268 A235873 A118743 * A333630 A114980 A376532

Adjacent sequences: A225223 A225224 A225225 * A225227 A225228 A225229

Keyword

nonn,fini,full

Author

Jonathan Sondow, May 06 2013

Status

approved