Arranging the Solutions of f(x+y+z) = xyz

Arranging the Solutions of f(x+y+z) = xyz

For any quadratic polynomial f(s) = as² + bs + c with integer coefficients a,b,c, consider the Diophantine equation f(x+y+z) = xyz. In other words, for fixed integers a,b,c we have the equation

If there exists a solution x,y,z to this equation, then z is a root of the quadratic

This can also be written in the form

Interestingly, we have z² = f(x+y)/a for a solution (x,y,z) if and only if (x,y) is a solution of f '(x+y) = xy.

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