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](http://books.google.com/books?id=7Vukkw8ywDgC&q=282#v=snippet&q=Theorem%252014.8&f=false) page 282, Theorem 14.8. We might also expect useful stuff in [Harman](http://press.princeton.edu/titles/8585.html). 


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](https://mathoverflow.net/questions/91945/).")


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$  

```


Primes $ 1 + x^2 + y^2$
It is known that there are infinitely many primes of this form; see the references in this previous thread [corrected link -- that'll teach me to post late at night!]:
[Primes represented by two-variable quadratic polynomials](https://mathoverflow.net/questions/55384/primes-represented-by-two-variable-quadratic-polynomials)


We still don't have an asymptotic for the number of primes $p \leq X$ of the form $1+x^2+y^2$, but the order of magnitude ($\asymp X/(\log{X})^{3/2}$) is known. 


