Search for sexy or octy prime

[Jens K Andersen]

http://www.lix.polytechnique.fr/Labo.../myprimes.html says Morain made the 10255-digit ECPP proof for the sexy prime in 60 cpu days. That sounds very fast for ECPP but is a slow way to find proven sexy primes. The ECPP certificate is 11.7 MB.
A week with other methods on my 2.4 GHz Core 2 Duo produced the 5 largest proven sexy primes and the 2 largest proven cousin primes, all with 11002 to 11004 digits. The expectation was only 1 sexy prime and 0.5 cousin prime.

Sexy primes:
(6929342093*((603077*8573#)^2-1)+2310)*603077*8573#/385 +1,+7
(15926901433*((603077*8573#)^2-1)+2310)*603077*8573#/385 -5,+1
(77244234818*((603077*8573#)^2-1)+2310)*603077*8573#/385 +1,+7
(85923872708*((603077*8573#)^2-1)+2310)*603077*8573#/385 +1,+7
(88450884888*((603077*8573#)^2-1)+2310)*603077*8573#/385 -5,+1

Cousin primes:
(8153374518*((603077*8573#)^2-1)+2310)*603077*8573#/385 +1,+5
(11799590168*((603077*8573#)^2-1)+2310)*603077*8573#/385 +1,+5

A twin prime was also found but it is far from the top-20:
(89163058273*((603077*8573#)^2-1)+2310)*603077*8573#/385 +/- 1

My APTreeSieve sieved. All prp tests were made by PrimeForm/GW (PFGW) which also made all proofs, except that 3 were completed by a Konyagin-Pomerance proof with a PARI/GP script by David Broadhurst from http://primes.utm.edu/primes/page.php?id=85564#comments.
The 16 proofs can be made in around 15 cpu minutes in total.
Documentation of the proofs is attached in proofs.zip.

The construction described here was not invented by me.
603077*8573#+/-1 is a 3665-digit twin prime found and proved for this search with PFGW. PFGW can prove that N is prime if N-1 or N+1 is 33.33% factored. If the factorization is between 30% and 33.33% then the PARI/GP script can complete the proof.

Let f(k) = ((385*k+58)*((603077*8573#)^2-1)+2310)*603077*8573#/385.

f(k)+d never has a factor <= 8573 for d = -11, -5, -1, 1, 5, 7
f(k)-6, f(k) and f(k+6) are each around 33.3% factored with this:
603077*8573#-1 divides f(k)-6.
8573#/385 divides f(k).
603077*8573#+1 divides f(k)+6.

The 6 d values were sieved together to 2.2*10^11.
f(k)+1 was prp tested when none of them had a factor.
If it was prp then other d values were prp tested.
If d = -5, -1, 1, 5, or 7 gives a prp N then there is 33.3% factorization of N-1 or N+1, and primality proofs are easy. Either PFGW can prove them by finding a couple of other small factors, or the PARI/GP script can complete the proof.
d = -1 was included in the sieve to give 3 instead of 1 small chances of a prime triplet: (-5, -1, 1), (-1, 1, 5), (1, 5, 7).
d = -11 would have required ECPP to prove but was included to give 2 instead of 1 small chances of a sexy triplet: (-11, -5, 1), (-5, 1, 7).

266 prp's of form f(k)+1 were found. 8 pairs for some d were found. No triplets were expected and none were found.

One of the reasons for the special construction of f(k) is to get a relatively short expression for the primes. If decimal expansions were acceptable instead then the Chinese remainder theorem could easily be used to contruct other functions g(k) with 3 arbitrary coprime numbers giving 33.3% factorization of g(k)-6, g(k) and g(k)+6.