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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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About the validity of a new conjecture about a diophantine equation

Let us consider the following conjecture: Conjecture: There are no integer solutions of the equation $$x^{y-z}z^{x-y}=y^{x-z}$$ with $x,y,z$ distinct positive integers greater than or equal to $2$. ...