I am currently using a non-systematic, pseudo-random method for finding prime-generating polynomials, based on the Bateman-Horn method for finding likely candidates, and then narrowing down. I have been so far, only searching for quadratics, searching up to a depth of only $10$ primes initially with values over $3$, and then refining. I have then been using the quadratic patterns from here with coefficient patterns $+,-,+$ as per the records published therein.

However, I have only been getting max $28$ prime-runs (aside from the expected longer runs associated with the Heegner numbers), searching $10^5$ interations, and then up to $10^6$ $n$ for each quadratic (which is no doubt the downfall, since the resultant runs appear to increase (as expected) the higher the search limit.

Is a more systematic approach more likely to reap better results, or is there a far more efficient method for searching?

eg MMA code thanks largely to @IgorRivin's answer here

Implemeted in stages (eg):

search2[-1, -1, -1]

Most@# & /@ %

With[{a = %}, {hhh[#] & /@ a, searchpt2[#] & /@ a}]

Flatten[Rest@# & /@ %[[2]]]

vv@# & /@ %