Given $n_1,n_2,m>0$ is there always a $t<\infty$ such that there are odd $a_1,\dots,a_{n_1}\in\{1,\dots,t\}$ and even $b_1,\dots,b_{n_2}\in\{1,\dots,t\}$ with each $a_i^2+b_j^2$ a distinct prime $>m$?
When such $t$ exists what is its minimum?
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The answer to your first question is yes. Indeed, you can even insist that $a_1,b_1,a_2,b_2,\dots$ forms an arithmetic progression (or any other predetermined constraints on the ratios of their gaps).
This follows from a theorem of Tao that the Gaussian primes contain "constellations" of arbitrary shape. As an example, there are infinitely many Gaussian integers $a$ and regular integers $r$ such that each of the Gaussian integers $a+r(2k-1+2\ell i)\colon 1\le k,\ell\le K$ is a Gaussian prime. One can avoid the regular primes $p\equiv3\pmod 4$ because they have relative density $0$ in the Gaussian primes, so the Gaussian primes $a+bi$ so found have the property that $a^2+b^2$ is a regular prime. I'm sure that the distinctness condition can be ensured through ad hoc reasoning.
answered Mar 17, 2018 at 20:18