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Oct 22, 2020 at 0:20 comment added Zhi-Wei Sun If $y\equiv-3x\pmod7$ and $z\equiv2x\pmod 7$, then $x+y+z\equiv 0\pmod 7$ and so your identity is helpful to Question 2. If $x^2\equiv y^2\equiv z^2\not\equiv0\pmod 7$, then $x+y+z\not\equiv0\pmod 7$ and hence the identity offers no help to Question 1.
Oct 22, 2020 at 0:07 history answered user125345 CC BY-SA 4.0