Cubic Diophantine equation
问题内容
Problem:I am looking for help with the following Diophantine equation:
$$y^2 = x^3 - x^2 + 16$$
By working through the equation, I have successfully found 8 distinct non- negative integer solutions. The largest value of $x$ among all the solutions I found is $x = 112$ (which gives $y = 1180$).
My question is simple: Is $112$ provably the absolute largest integer solution for $x$, or is there a bigger one hidden further out?
If $112$ is the definitive maximum, could someone explain or link to the method used to prove that no larger solutions can exist?
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