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  • $\begingroup$ Thank you for the proof of Conjecture 1. Note that $\prod_{1\le j<k\le m\atop p\nmid j^2+k^2}(j^2+k^2)\equiv (-1)^{\lfloor(p-5)/8\rfloor}\pmod p$ for any prime $p\equiv 1\pmod 4$, by (1.7) of my paper (arXiv:1809.07766). $\endgroup$ Commented Jul 7, 2019 at 21:45
  • $\begingroup$ Ah, indeed. By the way, Conjecture 2 may be proved in the same spirit, we need only to express the parity this number of residues in RHS via $2^{(p-1)/4}$ that is known (I guess belongs to Yamamoto). $\endgroup$ Commented Jul 7, 2019 at 22:17
  • $\begingroup$ Conjecture 3 also follows. We get the same additional multiple as in Conjecture 2, and it cancels out. It looks like for 32 we again get the same parity as in conjecture 2, for 64 again 0 and so on. $\endgroup$ Commented Jul 8, 2019 at 9:44