Need there be infinitely many Gaussian primes along lines that contain at least one? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T09:43:15Z http://mathoverflow.net/feeds/question/91692 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91692/need-there-be-infinitely-many-gaussian-primes-along-lines-that-contain-at-least-o Need there be infinitely many Gaussian primes along lines that contain at least one? Gray Taylor 2012-03-20T07:39:34Z 2012-03-20T11:22:04Z <p>Greetings from EuroCG 2012, from which I post via iPod, so apologies for lack of problem motivation, background research and mathematical formatting.</p> <p>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?</p> <p>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.</p> <p>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?</p> http://mathoverflow.net/questions/91692/need-there-be-infinitely-many-gaussian-primes-along-lines-that-contain-at-least-o/91701#91701 Answer by unknown (google) for Need there be infinitely many Gaussian primes along lines that contain at least one? unknown (google) 2012-03-20T09:57:49Z 2012-03-20T09:57:49Z <p>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?!</p>