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A system of nonlinear Diophantine equations whose positive solutions are not coprime

Consider the following system of Diophantine equations: $$v_1k_1=k_1^3-k_2^3+k_3^3 \\ v_2k_2=k_1^3+k_2^3-k_3^3 \\ v_3k_3=-k_1^3+k_2^3+k_3^3 \tag{1}$$ where $v_1,v_2,v_3$ and $k_1,k_2,k_3$ are integer-...
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