A084699 Composite integers j such that binomial(2*j,j) == 2^j (mod j).

12, 30, 56, 424, 992, 16256, 58288, 119984, 356992, 1194649, 9973504, 12327121, 13141696, 22891184, 67100672, 233850649

Offset: 1

Author

Benoit Cloitre, Oct 15 2003

Keywords

• nonn
• more

Comments

If p is prime, binomial(2*p,p) == 2^p (mod p).

a(17) > 10^9.

From Gabriel Guedes and Ricardo Machado, Nov 16 2023: (Start)

Theorem. Let j = (2^k)*p, where p is an odd prime and k is in N; then binomial(2*j,j) == 2^j (mod j) if and only if p satisfies the following conditions:

a) p divides binomial(2^(k+1),2^k) - 2^(2^k);

b) p has at least k 1's in its binary expansion.

Theorem. If m is an even perfect number then j = 2m satisfies the congruence binomial(2*j,j) == 2^j (mod j). See A000396.

Theorem. Let j = p^2 with p a prime number. Then p is a Wieferich prime if and only if binomial(2*j,j) == 2^j (mod j). See A001220. (End)

Contains 17179738112 and 274877382656 (from Guedes-Machado paper). - Michael De Vlieger, Nov 22 2023

Contains 3386741824, 750984028672, 33029195197184, 1145067923695616, 422612863956511744. - Ricardo Machado, Nov 23 2023

Contains 84385517065596416, 62648180117928433664, 273984397779878971648, 36506097537257040703232. - Max Alekseyev, Dec 07 2023

Links

• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems, sect. III: Binomial coefficients modulo integers, binomod.gp (V. 1.4, 11/2015).
• Gabriel Guedes and Ricardo Machado, Perfect numbers, Wieferich primes and the solutions of binomial(2n, n) congruent to 2^n mod n, Notes Num. Theor. Disc. Math. (NNTDM 2023) Vol. 29, No. 4, 705-712.

Crossrefs

Contains A139256 as a subsequence.

Cf. A015910, A059288, A080469, A082180, A109760, A109769, A228562, A328497.

Programs

PARI

lista(nn) = {forcomposite(n=1, nn, if (binomod(2*n, n, n) == Mod(2, n)^n, print1(n, ", ")));} \\ Michel Marcus, Dec 06 2013 and Dec 03 2023

Extensions

More terms from David Wasserman, Jan 03 2005

a(11)-a(16) from Max Alekseyev, Aug 05 2011