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
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
KEYWORD
nonn
AUTHOR
M. F. Hasler, Jan 06 2016
STATUS
approved