5
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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?!

$\endgroup$

Your Answer

By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.