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The primes in the ring of Gaussian integers $\mathbb Z[i]$ has a simple connection to the ordinary primes: either one of the components is zero and the other is a prime of type $4n+3$ or the norm $\|a+ib\|=a^2+b^2$ is a prime $a\ne 0\ne b$, that is a prime of the type $4n+1$. All $n\in\mathbb Z^+$ with $n=a^2+b^2$, e.g. all ordinary primes of the form $4n+1$, are composites in $\mathbb Z[i]$: $n=(a+ib)(a-ib)$.

There is essentially one pair of Gaussian primes such that $|z_1-z_2|=1$, and that is $(1+i,2+i)$. If defining prime pairs in $\mathbb Z[i]$ with $|z_1-z_2|=2$, then it seems like for each integer $n>0$ there is a prime $n+im\in\mathbb Z[i]$ with a prime as a pair. Can this be proved?

Verified for all $a+ib\in\mathbb Z[i]$ with $0<a,b\leq1000$.


I'll elaborate.

If $|z_1-z_2|=2$ one can assume e.g. that $\Re (z_1-z_2)=0$ and $|\Im (z_1-z_2)|=2$. Hence, the conjecture can be formulated as:

For each $a\in\mathbb N^+$ it exists $b\in\mathbb Z$ such that $a+ib$ and $a+i(b+2)$ are Gaussian primes.

Equivalent conjecture: $\forall n\in\mathbb N^+\exists m\in\mathbb Z: n^2+m^2,n^2+(m+2)^2\in\mathbb P$.

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    $\begingroup$ This is essentially an equivalent of the twin primes conjecture for $\mathbb{Z}[i]$, so I won't expect a simple proof, or just a proof, to appear soon. However, Terence Tao has proved many interesting facts about constellations of primes in $\mathbb{Z}[i]$: arxiv.org/abs/math/0501314 $\endgroup$ – Jack D'Aurizio Mar 23 '17 at 18:05
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This implies a particular case, with $(m_1,m_2)=(0,2)$, of Conjecture 1.2 in http://www.mast.queensu.ca/~akshaa/gaussian.pdf

The same paper in Theorem 1.4 proves: There are infinitely many rational primes of the form $p_1=a^2+b^2$ and $p_2=a^2+(b+h)^2$, with $a,b,h\in\Bbb{Z}$, such that $0<|h|\le 246$.

So if you can prove your conjecture, you can lower the bound from 246 to 2. The paper has appeared in Journal of Number Theory Volume 171, February 2017, Pages 449–473.

You would also solve the "frogger problem":

Conjecture: For any integer $a > 0$ there exists a rational prime of the form $x^2+a^2$ with integer $x$.

Found at page 26 of https://arxiv.org/pdf/1606.05971.pdf

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