I am currently using a non-systematic, pseudo-random method for finding prime-generating polynomials, based on the [Bateman-Horn method](https://dl.dropboxusercontent.com/u/5188175/BatemanHorn.pdf) 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](http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html) 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](https://github.com/martinq321/primepoly/blob/master/mma) *thanks largely to @IgorRivin's answer [here](https://mathoverflow.net/a/214726/45057)*

*Implemeted in stages (eg):*

    search2[-1, -1, -1]

    Most@# & /@ %

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

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

    vv@# & /@ %