Motivated by Mathoverflow question here I have recently became interested in solving Polynomial Diophantine equations, that is, equations of the form
for some polynomial P with integer coefficients. Because there are many such equations, I have decided to ask a computer to help me. Our conversation is presented here.
Highlights of the conversation: the height of a polynomial over Z in many variables is what you get when you make all of the coefficients positive and plug in x=2. For example, the height of
Note that, for all h, there are only a finite number of polynomials in many vars over Z with height h. With the help of my friend the computer we have looked at the equations with h=0,1,2,... and so on, and tried to determine which ones have any integer solutions. As expected, the first equations were trivial, but at about h=22 we have started to meet quite interesting equations for which we needed help from Mathoverflow to solve. The project is currently at h=29, with only one remaining open equation of this height. Read the conversation with the computer, or my mathoverflow question