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Suppose we consider primes of the form $p = 1 \text { mod } 4$, so that $p = a^2 + b^2$, $a$ and $b$ being integers. Considering only the first quadrant, all $(a,b)$ pairs will be of the form (odd,even) or (even,odd).

Now if we consider $q = (a + n)^2 + (b + n)^2 = p + 2na + 2nb + 2n^2$, $n$ an integer, then $q = 1 \text { mod } 4$.

$1 \text { mod } 4$ as an arithmetic progression contains an infinite number of primes, and while every $q$ may not turn out to be a prime, one would still expect an infinite number of prime pairs $(p,q)$.

Has any work been done related to this? My original motivation was that this could give very close Gaussian primes, differing by a distance of just $n\sqrt2$, when $n$ is small.

Update: I came across a recent paper (Bounded gaps between Gaussian primes) where instead of $(a + n,b + n)$, the author considers $(a + n,b + m)$, with $(n,m)$ either $\text (odd,odd)$ or $\text (even,even)$, and then proves it for the case $(0,m)$, $m>=246$.

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    $\begingroup$ Check out Bunyakovsky conjecture. $\endgroup$ – Wojowu Dec 26 '15 at 15:24
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    $\begingroup$ This is a particular case of a natural conjecture that would generalize Dirichlet's theorem to the Gaussian integers, but this generalization is not known. I doubt this particular case is known. It is a famous open problem whether there are infinitely many primes in another arithmetic progression, $n+i$, or equivalently, infinitely many primes in the integers of the form $n^2+1$. $\endgroup$ – Douglas Zare Dec 26 '15 at 15:26
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    $\begingroup$ I am pretty sure that Bunyakovsky conjecture is open for any specific nonlinear polynomial. And probably (less sure here) it is not even proved that there exists a nonlinear polynomial for which it holds. $\endgroup$ – Fedor Petrov Dec 26 '15 at 15:54
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    $\begingroup$ Friedlander and Iwaniec's paper (arxiv.org/pdf/math/9811185.pdf) mentions that the problem in one variable is generally more difficult than that in two. So it's possible this problem is more tractable than Bunyakovsky's. $\endgroup$ – valar123 Dec 26 '15 at 18:05
  • $\begingroup$ @valar123 this question is equivalent to the specific case of Bunyakovsky conjecture, is not it? $\endgroup$ – Fedor Petrov Dec 27 '15 at 14:54
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An infinite number of primes on a ray of slope $45^{\circ}$ would not imply close Gaussian primes any more than ordinary primes $0^{\circ}.$ I didn't read the paper you link, but $246$ is the current record for gaps between ordinary primes. You might enjoy reading about Guassian moats.

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  • $\begingroup$ The result for ordinary primes by Maynard, Tao and Polymath was in 2013-14, while the paper I linked seems to be very recent (Nov 2015). I found the link on the Polymath project page (michaelnielsen.org/polymath1/…), so it seems credible. The author has adapted the original machinery for Gaussian primes. Maybe the first question doesn't have a elementary solution. I read about Gaussian moats, and the subject is indeed fascinating :) $\endgroup$ – valar123 Dec 27 '15 at 19:54

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