Greetings from EuroCG 2012, from which I post via iPod, so apologies for lack of problem motivation, background research and mathematical formatting.

Question:Suppose L is a horizontal or vertical line in the argand plane passing through a Gaussian prime. Are there infinitely many Gaussian primes on L?

In fact, all I need is a next prime along a line, but of course if that was guaranteed one could repeat the process to keep going forever. Still, if there is a next prime, some idea of how far along it is might also be useful for the application in mind.

Hopefully equivalent question for rational primes in rational integer sequences: let s(k)=a^2+(b+k)^2 for k>=0. if s(0) is prime, does the sequence {s(k)} contain infinitely many primes?