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Making use of induction. Suppose [itex]a,b,c[/itex] is a nonzero solution. Due to [itex]x^2\equiv 0,1\pmod{3}[/itex] it must hold that [itex]a^2,b^2[/itex] are multiples of three. Putting [itex]d:= (a,b)[/itex], if [itex]d\mid c[/itex] holds, then [itex]\left (\frac{a}{d},\frac{b}{d},\frac{c}{d}\right )=:(a',b',c')[/itex] would be a solution with [itex](a',b')=1[/itex] which is impossible.
Could we show [itex]d\mid c[/itex]?
Could we show [itex]d\mid c[/itex]?