Consider the following quadratic equation where $a, b > 0$ are integers:
- $x_2^2 = x_1^2 - ax_1 + b^2$
I am looking for the rational solutions of the above equation satisfying the following conditions:
$0 < x_1 < \frac{a}{2}$
$\sqrt{b^2 - \frac{a^2}{4}} < x_2 < b$
$2b > a$
Now consider the following conjectures:
Conjecture 1: There are infinitely many rational values of $x_1, x_2$ satisfying the above equation for each possible positive integer values of $a, b$.
Conjecture 2: If the Conjecture 1 is true, for a given integers $a, b$, the rational values of $x_1, x_2$ satisfying the above equation form a dense set.
Questions:
Are the above conjectures true ? Are there any similar known conjectures or results ?
If the above conjectures are false, what are the conditions (necessary and/or sufficient) on the integer values of $a, b$ such that the above conjectures are true.
Write your original equation as:
$$\left(x_1-\frac{a}{2}\right)^2-x_2^2=\frac{a^2}{4}-b^2$$
$$\left(x_1-\frac{a}{2}-x_2\right)\left(x_1-\frac{a}{2}+x_2\right)=\frac{a^2}{4}-b^2$$
... so you could equate, for example $x_1-x_2-\frac{a}{2}=1$, $x_1+x_2-\frac{a}{2}=\frac{a^2}{4}-b^2$ and get a rational solution for $x_1$ and $x_2$.
So the answer to your question is yes, there is an infinite number of solutions for each choice of integers $a$ and $b$.