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See Conjecture 4.8 of my paper available from http://maths.nju.edu.cn/~zwsun/192d.pdf
The result for the case $p\equiv3\pmod4$ is not new, see http://maths.nju.edu.cn/~zwsun/202d.pdf for a proof of this and some other similar results.
The result is not new, see http://maths.nju.edu.cn/~zwsun/202d.pdf for a proof of this and some other similar results.
