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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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    $\begingroup$ 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 $c$. $\endgroup$
    – Noam D. Elkies
    Commented Mar 24, 2012 at 4:48
  • $\begingroup$ @Noam, thanks. I'm not seeing any references on this. $\endgroup$
    – Will Jagy
    Commented Mar 24, 2012 at 6:20
  • $\begingroup$ @Will: I added a link to the prime-spiral question to which you obliquely refer. $\endgroup$
    – Joseph O'Rourke
    Commented Mar 24, 2012 at 9:29
  • $\begingroup$ (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.) $\endgroup$
    – Noam D. Elkies
    Commented Mar 24, 2012 at 15:27
  • $\begingroup$ @Joseph, It's twue, it's twue! imdb.com/title/tt0071230/quotes $\endgroup$
    – Will Jagy
    Commented Mar 24, 2012 at 21:33

1 Answer 1

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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

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.

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    $\begingroup$ The link you give is to this same question... Nice to know this has an answer, though. $\endgroup$
    – Will Jagy
    Commented Mar 24, 2012 at 7:06
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    $\begingroup$ I'm sure anonymous will update the link he offered, but in the meantime another reference is Friedlander and Iwaniec's Opera de Cribro, Theorem 14.8. I'm not sure who proved the result in the first place. $\endgroup$
    – Brad Rodgers
    Commented Mar 24, 2012 at 8:15
  • $\begingroup$ @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 $\endgroup$
    – Will Jagy
    Commented Mar 24, 2012 at 17:39
  • $\begingroup$ Thanks. Theorem 2 on journal page 436 of Iwaniec (1974) gives all that I was asking. $\endgroup$
    – Will Jagy
    Commented Mar 24, 2012 at 21:12

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