I have a related question to one I've read today, see: Integer solutions to $2x^2+5x+y^2=19$
The integers solution are part of an ellipse, with an obvious finite number of $x$. What I would like to know is, when we have a large ellipse so that computing each $x$ would be disastrous by hand, is there any method to cut the possibilities down, or provide an exact answer for an ellipse? Or are there only approximation algorithms that can accomplish this?
If $x$ and $y$ are positive integers you can write: $$5x+y^2\equiv 1\pmod 2$$ But the acceptable values of $x$ are $1,2,3$ because $5x<19$ while the acceptable values of $y$ are $1,2,3,4$ because $y^2<19$. You can consider two systems: $$ \left\{ \begin{array}{c} 5x\equiv 0\pmod 2\\ y^2\equiv 1\pmod 2 \end{array} \right. $$ And $$ \left\{ \begin{array}{c} 5x\equiv 1\pmod 2\\ y^2\equiv 0\pmod 2 \end{array} \right. $$ In the first systems the acceptable value of $x$ is only $2$ and the acceptable values of $y$ are $1,3$. In the second system the acceptable values of $x$ are $1,3$ while $y$ are $2,4$. Now you can control the solitions. Edit: to find also the negative values you can consider the equation of second degree in $x$: $$2x^2+5x+y^2-19=0$$ The discriminant of equation is $\Delta=177-8y^2$ but $$177-8y^2\ge 0$$ therefore the accetable values of $y$ are:${{-4,-3,....,3,4}}$ but $$177-8y^2=k^2$$ and the only values that solve the equation are $y=$ $\pm 1$ and $\pm 4$.