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In April 2018, I noted that the first integer $n>1$ with $n^2\not\in\{x^2+2y^2+3\times 2^z:\ x,y,z=0,1,2,\ldots\}$ is $$5884015571=7\times17\times49445509.$$

Question. Is it true that for each prime $p$ we can write $p^2$ as $x^2+2y^2+3\times2^z$ with $x,y,z$ nonnegative integers?

I guess that this question has a positive answer. Your comments are welcome!

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  • $\begingroup$ Out of curiosity, in what context did you discover this fact? $\endgroup$
    – user147820
    Commented Apr 7, 2020 at 1:51
  • $\begingroup$ I think your form is suitable only for 2 to 29 , Try p=31 $\endgroup$
    – zeraoulia rafik
    Commented Apr 7, 2020 at 2:55
  • $\begingroup$ @zeraouliarafik $31^2 = 29^2 + 2 \cdot 6^2 + 3 \cdot 2^4 $. $\endgroup$
    – Robert Israel
    Commented Apr 7, 2020 at 3:46
  • $\begingroup$ OP put a related sequence online, oeis.org/A301472 $\endgroup$
    – Gerry Myerson
    Commented Apr 7, 2020 at 4:42
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    $\begingroup$ The density of the 3-variable sequence seems fine to me but the question still feels like a smooth glass mountain. Do you see at this time any chance of climbing or even starting it? $\endgroup$
    – Wlod AA
    Commented Apr 7, 2020 at 6:26

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