In this note, I will discuss how to find the maximum possible integer solutions for x and y in the slight variant of the quadratic Diophantine equation . We will focus on the case where since we can factor the RHS into and every integer is a solution for .
Begin with the generic quadratic .
Since is guaranteed a perfect square, must be a difference of squares. A short proof for this is
This means that the left side can be rearranged to
NOTE: is a variable that is an integer.
This is maximized when the denominator is minimized or as the numerator tends infinity. Thus to minimize the denominator, (look familiar? It's the same as the negation of the vertex of a parabola. Why this is, I have no idea). The sign (it means approximately) is important for two reasons. 1) because has to be integral for y to be integral and isn't always integral. 2) can't exactly equal because if it does, then we are dividing by 0 and the world will implode.
NOTE: this is a plug and chug because we need to find an integral value of n that will make the numerator divisible by the denominator.
Now that we have found the maximum value of x, we can plug this in to find y.
(the long bars denote absolute value.. sorry it looks so bad)
This also further proves that y is maximized when is as close as possible to .