EDIT, Saturday 11:32 am, March 24: a complete answer for $4 + x^2 + y^2$ and generally $4 m^2 + x^2 + y^2$ for fixed $m$ is given in Friedlander and Iwaniec page 282, Theorem 14.8. We might also expect useful stuff in Harman.

ORIGINAL: There is no inspiration involved with this. It just happened. Well, it is about a possible method of giving Joseph some prime spirals. For his birthday. (See "Primes that are the sum of three squares.")

I do not seem to know whether there are infinitely many primes of the form $1 + x^2 + y^2,$ for that matter $4 + x^2 + y^2,$ or $9 + x^2 + y^2,$ or $16 + x^2 + y^2.$ So that is the question, for fixed $a,$ are there infinitely many primes $a^2 + x^2 + y^2?$

It seems a very good bet. Every prime $p$ not with $p \neq 7 \pmod 8$ can be expressed as $x^2 + y^2 + z^2.$ Furthermore, we know that all $p \equiv 1 \pmod 4$ can be written as $x^2 + y^2,$ so that is one example of an infinite set.

Furthermoremore, the count of numbers up to some large $N > 0$ that can be written as $a^2 + x^2 + y^2$ is asymptotically the same as the result with $a=0,$ namely $$ \frac{0.7642... \; \; N \;}{\sqrt{\log N}} $$

Looking at some computer output, it does seem to be very easy to write a bunch of primes with fixed $a$ in $a^2 + x^2 + y^2.$

```
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./primes_1_x2_y2 | sort -n
2 1 0 1
3 1 1 1
5 1 0 2
11 1 1 3
17 1 0 4
19 1 3 3
37 1 0 6
41 1 2 6
53 1 4 6
59 1 3 7
73 1 6 6
83 1 1 9
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./primes_1_x2_y2 | sort -n
5 2 0 1
13 2 0 3
17 2 2 3
29 2 0 5
29 2 3 4
41 2 1 6
53 2 0 7
89 2 2 9
89 2 6 7
101 2 4 9
113 2 3 10
149 2 1 12
149 2 8 9
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./primes_1_x2_y2 | sort -n
11 3 1 1
13 3 0 2
17 3 2 2
19 3 1 3
29 3 2 4
41 3 4 4
43 3 3 5
59 3 1 7
59 3 5 5
61 3 4 6
67 3 3 7
73 3 0 8
83 3 5 7
89 3 4 8
107 3 7 7
109 3 0 10
109 3 6 8
113 3 2 10
131 3 1 11
137 3 8 8
139 3 3 11
139 3 7 9
157 3 2 12
173 3 8 10
179 3 1 13
179 3 7 11
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./primes_1_x2_y2 | sort -n
17 4 0 1
29 4 2 3
41 4 0 5
41 4 3 4
53 4 1 6
61 4 3 6
89 4 3 8
97 4 0 9
101 4 2 9
101 4 6 7
113 4 4 9
137 4 0 11
173 4 6 11
197 4 9 10
241 4 0 15
241 4 9 12
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$
```