Question

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 $k\ge0$. If $s(0)$ is prime, does the sequence $\{s(k)\}$ contain infinitely many primes?

Answer 1

There is the Hardy-Littlewood Conjecture F and the Bateman-Horn conjecture. But for more refined treatment on these Gaussian prime gaps (analogously, gaps in numbers mapping to primes represented by irreducible polynomials $f$, gaps between principal prime ideal generators along lines through algebraic number fields embedded in the right dimension), the question we really need to ask is, is there also a "Cramér model", something that expresses the gaps between $n$ and $n^{\prime}$, where $f(n), f(n^{\prime})\in \mathbb{P}:=$ set of primes, and $f(n^{\prime})$ is the next prime in the sequence of primes represented by $f$ after $f(n)$, in terms of a probability distribution?!