Integer Square Root Quadratics

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 $y^2=a^2x^2+bx+c$ . We will focus on the case where $b\neq 2a\sqrt{c}$ since we can factor the RHS into $(ax+c)^2$ and every integer is a solution for $x$ .

Begin with the generic quadratic $y=\sqrt{a^2x^2+bx+c}$ .

Since $a^2x^2$ is guaranteed a perfect square, $bx+c$ must be a difference of squares. A short proof for this is $y^2=a^2x^2+bx+c\Rightarrow y^2-a^2x^2=bx+c$

This means that the left side can be rearranged to

$\begin{aligned} ax+n &=& \sqrt{a^2x^2+bx+c}\\ a^2x^2+2axn+n^2 &=& a^2x^2+bx+c\\ bx-2axn&=&n^2-c\\ x(b-2an)&=&n^2-c\\ x&=&\boxed{\dfrac{n^2-c}{b-2an}}\\ \end{aligned}$

NOTE: $n$ 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, $n\approx \dfrac{b}{2a}$ (look familiar? It's the same as the negation of the vertex of a parabola. Why this is, I have no idea). The $\approx$ sign (it means approximately) is important for two reasons. 1) because $n$ has to be integral for y to be integral and $\dfrac{b}{2a}$ isn't always integral. 2) $n$ can't exactly equal $\dfrac{b}{2a}$ 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.

$y=\sqrt{a^2x^2+bx+c}$

$y=\sqrt{a^2\left(\dfrac{n^2-c}{b-2an}\right)+b\left(\dfrac{n^2-c}{b-2an}\right)+c}$

$rearranging...$

$\boxed{y=\left|\dfrac{bn-a(n^2+c)}{b-2an}\right|}$

(the long bars denote absolute value.. sorry it looks so bad)

This also further proves that y is maximized when $n$ is as close as possible to $\dfrac{b}{2a}$ .

Practice section

Maximum possible integer square root polynomial 1 (easier)

Maximum possible integer square root polynomial 2 (harder)

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Integer Square Root Quadratics

Practice section

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