Is it true that $8k+3=x^2+y^2+p^2,\forall k\in \mathbb N$? Is it true that for every $k\in \mathbb N,$ we can find $x,y\in \mathbb Z$ and $p=1$ or $p\in P $ such that $$8k+3=x^2+y^2+p^2$$? I checked this for all $k\leq 10^5,$ but cannot prove it.
Thanks in advance!

This is apparently a hard problem. Duke, Friedlander, and Iwaniec showed in their Int. Math. Res. Notices paper "Weyl sums for quadratic roots" that this is true for all sufficiently large $k$, but they had to assume two big conjectures in analytic number theory (GRH and Elliott-Halberstam). See Theorem 1.6, although there it's stated for different congruence conditions than the $N\equiv3\pmod 8$ you're interested in.
