A286230 Possible differences between consecutive primes p, q satisfying p! == 1 (mod q).

2, 12, 30, 76

Offset: 1

Author

Max Alekseyev, May 04 2017

Keywords

• bref
• nonn
• more

Comments

By Wilson's theorem, p! == 1 (mod p+2) whenever p, p+2 are twin primes, so 2 is a term.

Terms d > 2 correspond to Wilson's pseudo-twin primes, i.e., consecutive primes p, q such that q - p = d and p! == 1 (mod q). The corresponding primes are listed in A286181 and A286208.

As a set, the sequence is the union of {2} and the differences {A286208(n) - A286181(n)}.

A positive integer d belongs to this sequence if A033312(d-1) = (d-1)! - 1 has a prime factor q such that q - A151799(q) = d.

All terms are even.

If it exists, a(5) >= 140.

Crossrefs

Cf. A275111, A286181, A286208.