nonlinear Diophantine quadratic equations

Question

given the following

$f(x) = x^2 - 1068 x$.

$f(y) = y^2 - 926 y + 314$.

is there a mathematical method to find solution

Answer 1

I think you mean $x^2 - 1068 x = y^2 - 926 y + 314$. With $x = X + 534$ and $y = Y + 463$ this simplifies to $X^2 - Y^2 = 71101$. Since $X^2 - Y^2 = (X+Y)(X-Y)$ and $71101$ is odd, solutions will correspond to divisors of $71101$: given integer $a$ such that $a \mid 71101$, take $b = 71101/a$ (note both $a$ and $b$ are odd), and $X = (a+b)/2$, $Y=(a-b)/2$ is a solution.