OFFSET
1,1
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.
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paolo Iachia, Mar 25 2019
STATUS
approved