# About the distribution of Gaussian prime pairs

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

• 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 – Jack D'Aurizio Mar 23 '17 at 18:05

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 $0<|h|\le 246$.
Conjecture: For any integer $a > 0$ there exists a rational prime of the form $x^2+a^2$ with integer $x$.