Edited by _Don Reble (djr(AT)nk.ca), _, Jun 11 2007

_Cino Hilliard (hillcino368(AT)gmail.com), _, May 02 2005

First a(n) = the smallest prime p(n)-8 in the progression where such that p(n)-+8 and p(nq+k)-8 are both prime k=1,2.; q is the n-th prime after p.

3, 5, 3, 11, 5, 3, 3, 23, 89, 11, 11, 5, 3, 3, 59, 29, 11, 11, 5, 3, 23, 5, 3, 11, 29, 5, 3, 29, 11, 59, 71, 5, 3, 5, 3, 29, 5, 3, 89, 23, 11, 233, 5, 3, 23, 5, 3, 23, 11, 71, 5, 3, 59, 5, 3, 23, 173, 149, 101, 131, 11, 29, 23, 5, 3, 101, 479, 11, 11, 5, 3, 3, 53, 71, 233, 71, 59, 53, 29, 23, 71, 101, 173, 191, 59, 11, 29, 5, 3, 23, 11, 71, 5, 3, 5, 3, 29, 53, 11, 263, 29, 5, 3

Conjecture 1: There are an infinite number of primes p(n)-8 such that p(n)-8 and p(n+k)-8 are both prime for all k=1,2,3... Also, these numbers are components of octal primes. Generally, Conjecture 2: There are an infinite number of primes p(n)-2m such that p(n)-2m and p(n+k)-2m are both prime for all k>0, 2m > 4. Also, these numbers are components of primes that differ by 2m.

a(2)=5 because 5+8 and 11+8 are primes.

(PARI) pnpklist(n, m) = \ both are prime { local(x, l1, l2, v1, v2); for(k=1, n, for(x=1, n, v1 = prime(x)+ m; v2 = prime(x+k)+m; if(isprime(v1)&isprime(v2), \ print1(x", ") print1(v1", "); break; ) ) ) }

Cf. A105439, A105443, A105444.

uned,obsc,nonn,new

nonn

Edited by Don Reble (djr(AT)nk.ca), Jun 11 2007

uned,obsc,nonn,new

Cino Hilliard (hillcino368(AT)hotmailgmail.com), May 02 2005

First prime p(n)-8 in the progression where p(n)-8 and p(n+k)-8 are both prime k=1,2..

3, 5, 3, 11, 5, 3, 3, 23, 89, 11, 11, 5, 3, 3, 59, 5, 3, 23, 5, 3, 11, 29, 5, 3, 29, 11, 59, 5, 3, 29, 5, 3, 89, 11, 23, 5, 3, 11, 59, 5, 3, 23, 149, 101, 11, 29, 5, 3, 101, 11, 11, 5, 3, 3, 53, 71, 233, 71, 59, 53, 29, 23, 71, 101, 11, 29, 5, 3, 23, 11, 71, 5, 3, 5, 3, 29, 11, 263, 5, 3

1,1

Conjecture 1: There are an infinite number of primes p(n)-8 such that p(n)-8 and p(n+k)-8 are both prime for all k=1,2,3... Also, these numbers are components of octal primes. Generally, Conjecture 2: There are an infinite number of primes p(n)-2m such that p(n)-2m and p(n+k)-2m are both prime for all k>0, 2m > 4. Also, these numbers are components of primes that differ by 2m.

(PARI) pnpklist(n, m) = \ both are prime { local(x, l1, l2, v1, v2); for(k=1, n, for(x=1, n, v1 = prime(x)+ m; v2 = prime(x+k)+m; if(isprime(v1)&isprime(v2), \ print1(x", ") print1(v1", "); break; ) ) ) }

uned,obsc,nonn

Cino Hilliard (hillcino368(AT)hotmail.com), May 02 2005

approved