Solving a diophantine equation

Question

Given the following function: $$f(x) = \sqrt{ (2155 - 6x)^2-4x}$$ where x is an integer and the function also generates an integer value, is there an algorithm to determine its integer solutions?

Answer 1

HINT : If you want to find $(x,y)\in\mathbb Z$ such that $$y^2=(2155-6x)^2-4x,$$ since we have $$36 x^2-25864 x+4644025-y^2=0\Rightarrow x=\frac{6466\pm\sqrt{9y^2+12931}}{18},$$ we know that $9y^2+12931$ has to be a perfect square.

Hence, for some $z\in\mathbb Z$, we have $$9y^2+12931=z^2\iff (z-3y)(z+3y)=67\times 193$$ where $67,193$ are primes. Then, you can find $(y,z)$ to get $(x,y)$.