# Primes $1 + x^2 + y^2$

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$./primes_1_x2_y2 | sort -n 17 4 0 1 29 4 2 3 41 4 0 5 41 4 3 4 53 4 1 6 61 4 3 6 89 4 3 8 97 4 0 9 101 4 2 9 101 4 6 7 113 4 4 9 137 4 0 11 173 4 6 11 197 4 9 10 241 4 0 15 241 4 9 12 jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$

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We even expect infinitely many primes $a^2+x^2$ for any nonzero $a$ (indeed infinitely many $p=c+x^2$ for ixed nonzero $c$), so $p=a^2+x^2+y^2$should certainly be true. It's probably still very hard to prove, though I wouldn't be entirely surprised if it's known or within reach of current analytic technology. Meanwhile, as usual sieve estimates provide an upper bound that's of the same form $cN/\sqrt{\log N}$ as the expected asymptotic but with a larger $c$. – Noam D. Elkies Mar 24 '12 at 4:48
@Noam, thanks. I'm not seeing any references on this. – Will Jagy Mar 24 '12 at 6:20
@Will: I added a link to the prime-spiral question to which you obliquely refer. – Joseph O'Rourke Mar 24 '12 at 9:29
(Yes, I should have written $cN/\log^{3/2} N$, not $cN/\log^{1/2}N$, because there's already a $\log N$ in the denominator from the prime number theorem.) – Noam D. Elkies Mar 24 '12 at 15:27
@Joseph, It's twue, it's twue! imdb.com/title/tt0071230/quotes – Will Jagy Mar 24 '12 at 21:33

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.
@Brad, thanks. Page 282 at books.google.com/… For example, $\alpha = 1, \beta = -4,$ we get infinitely many primes $4 + x^2 + y^2.$ Another book with some evident key words, press.princeton.edu/titles/8585.html Glyn Harman Prime Detecting Sieves – Will Jagy Mar 24 '12 at 17:39