For instance, they are points in the geometric object known as the scheme over the integers.[1] The Riemann Zeta function of the Riemann Hypothesis generalizes to other schemes, and reflects their geometric properties (being vague because it's over a decade since I studied this stuff.) It is just barely in the realms of possibility that a new computationally simple way of finding large primes would also reflect something about the geometric structure of schemes over Z. (But it seems pretty damn unlikely to me.)

[1] http://en.wikipedia.org/wiki/Scheme_(mathematics)#Modern_def...