A266948 Least prime p such that p-2 and 6n-p are also prime, or 0 if no such prime exists.

0, 0, 5, 5, 5, 7, 5, 5, 5, 7, 7, 5, 5, 5, 5, 7, 7, 5, 5, 5, 7, 13, 5, 7, 5, 13, 5, 5, 5, 7, 7, 5, 13, 5, 5, 13, 5, 31, 5, 5, 7, 5, 13, 7, 7, 7, 5, 5, 5, 13, 7, 13, 5, 5, 7, 13, 5, 5, 31, 5, 7, 7, 5, 5, 5, 7, 7, 5, 7, 5, 19, 5, 13, 5, 5, 7, 7, 5, 5, 7, 13, 7, 5, 7, 5, 7, 7, 13, 5, 13, 19, 5, 5, 109, 7, 7, 5, 5, 19, 7, 7, 5, 5, 5, 5, 13, 5, 43, 5, 7, 7, 5, 13, 5, 7, 7, 5, 19, 7, 5, 19

Offset

0,3

Comments

Goldbach conjecture related: Group the consecutive even numbers in groups of three, (6n-2, 6n, 6n+2). The existence of a(n) corresponds to a Goldbach decomposition 6n = p + (6n-p) using the upper of a twin prime pair. Then 6n-2 = (p-2) + 6n-p is automatically a valid Goldbach decomposition of 6n-2, and 6n+2 = p + 6n+2-p is such a decomposition for 6n+2 if 6n+2-p (or 6n+4-p) is prime.

Zwillinger conjectured already in 1978 that for all n > 701 there is a p such that all these conditions are satisfied (not necessarily p = a(n)). See also A266952 - A266953.

This conjecture implies that a(n) > 0 for all n > 1.

See A266950 - A266951 for record values and indices. For easier reference we list some of these [n, a(n)] here: [21, 13]; [133, 139]; [1759, 241]; [10919, 643], [112723, 1621]; [1072318, 2311], [1458993, 3001], [2617393, 3301], ...

Since a larger value of a(n) indicates that it was "difficult" to find a suitable twin prime p, this slow growth is a strong evidence that a(n) > 0 for all n > 1.

Links

Table of n, a(n) for n=0..120.

Harvey Dubner, Twin Prime Conjectures, Journal of Recreational Mathematics, Vol. 30 (3), 1999-2000.

Bill Krys, Not much response but I still think this is outrageous result, Yahoo! group primenumbers, Jan. 6, 2016

Bill Krys, not much response, but i still think this is outrageous result, message in primenumbers Yahoo group, Jan 6, 2016 [cached copy].

Dan Zwillinger, A Goldbach Conjecture Using Twin Primes, Math. Comp. 33, No.147 (1979), p.1071.

Prog

(PARI) A266948(n)=my(GP(n, p=2)=forprime(p=p, n, isprime(n*2-p)&&return(p))); for(p=1, 3*n, isprime(-2+p=GP(3*n, p))&&return(p))

Crossrefs

Sequence in context: A159002 A127934 A205236 * A176172 A204911 A087516

Adjacent sequences: A266945 A266946 A266947 * A266949 A266950 A266951

Keyword

nonn

Author

M. F. Hasler, Jan 06 2016

Status

approved