Recursive equations for quadratic diophantine equations.

Question

How do I determine the recursive equations for deriving integer solutions to quadratic Diophantine equations. Say I have a Diophantine equation. e.g. $$5x^2-3x+3y^2+2y=0$$ How do I determine a recursive equation so that with one integer solution I can determine the other integer solutions.

Answer 1

I don't know what you mean by "recursive" in this case, but you might be interested in the solution that I posted here for generating all rational solutions using one guessed solution. The idea is that the slope of a line through the guessed point is rational if and only if the other point at which it intersects the curve is a rational point. If you are looking for integers solutions, you can find all rational solutions and then look into when the coordinates are integers, say by using divisibility theorems. For example, there is a well-known way of finding all (primitive) Pythagorean triples by using this method to first find all rational solutions to $x^2+y^2=1.$

However, if you are looking for integer solutions, and if the coefficient of $xy$ in the curve is $0$ and both the $x^2$ and $y^2$ terms exist, as in the example that you gave, there might be a different approach available: complete the square in both variables and look at whether you can take advantage of size considerations. For example, completing the square in your example $$5x^2-3x+3y^2+2y=0$$ and clearing the denominators yields $$3(10x-3)^2+20(3y+1)^2=47.$$ If you look at $$3a^2+20y^2=47,$$ it is clear that $b$ can only be $0,\pm 1$ before becoming too large. If $b=0,$ we get irrational $a.$ If $b=\pm 1,$ we get $a=\pm 3.$ We end up getting that the only integer solution is $(x,y)=(0,0).$