A306400 For the n-th prime p of the form 6k-1, a(n) is the first prime q for which (p+q^2-1, p+q^2+1) is a twin prime pair.

5, 7, 5, 7, 11, 23, 5, 7, 7, 11, 5, 7, 7, 11, 5, 7, 701, 7, 5, 5, 7, 7, 41, 11, 7, 19, 13, 5, 7, 31, 17, 13, 11, 31, 41, 13, 31, 7, 29, 11, 37, 13, 53, 11, 19, 19, 11, 13, 23, 37, 7, 41, 23, 7, 29, 5, 71, 5, 13, 29, 13, 13, 59, 97, 11, 37, 17, 7, 7, 5, 7, 157

Offset: 1

Author

Paolo Iachia, Mar 25 2019

Keywords

• nonn
• easy

Comments

Actually, there is no need to test for q=2 and q=3, as 6k-1+4+-1 = (6k+2, 6k+4), both terms not prime; and 6k-1+9+-1 = (6k+7,6k+9), with 6k+9 not prime.

The sequence could be extended to nonprime numbers p=6k-1 and/or nonprime numbers q=6t+-1. However, it could not be extended to p=6k+1 (prime or not), because for q=6t+-1, p+q^2 = 6k+1+36t^2+-12t+1 ≡ 2 (mod 6); hence p+q^2+1 == 3 (mod 6) is never a prime number.

Proving that this sequence is infinite would prove the twin prime conjecture (that there are infinitely many twin primes), as the twin prime pair associated with the prime p is greater than p and there are infinitely many prime numbers.

This sequence refers to the first q for which p+q^2+-1 is a twin prime pair. However, analysis (by computer program) suggests that for each prime p there are infinitely many primes q for which p+q^2+-1 is a twin prime pair. Proving this statement, even for a single prime p, would prove the twin prime conjecture.

Examples

			Example: for n=5, a(5) = 11, because the 5th prime of the form 6k-1 is 29, and 29+11^2+-1 = (149,151) is a twin prime pair, while 29+2^2+-1, 29+3^2+-1, 29+5^2+-1, 29+7^2+-1 are not twin prime pairs.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007528, A001359 (lesser of twin primes).

Programs

Maple

g:= proc(p) local q;
  q:= 3:
  do
    q:= nextprime(q);
    if isprime(p+q^2-1) and isprime(p+q^2+1) then return q fi;
  od
end proc:
map(g, select(isprime, [seq(i,i=5..1000,6)])); # Robert Israel, Nov 23 2020

Mathematica

Table[Block[{q = 2}, While[! AllTrue[p + q^2 + {-1, 1}, PrimeQ], q = NextPrime@ q]; q], {p, Select[Range[5, 825, 6], PrimeQ]}] (* Michael De Vlieger, Mar 31 2019 *)

PARI

lista(nn) = {forprime(p=2, nn, if (((p+1) % 6) == 0, my(q=5); while (!(isprime(p+q^2-1) && isprime(p+q^2+1)), q = nextprime(q+1)); print1(q, ", ");););} \\ Michel Marcus, Mar 26 2019