- Find a parametrization (similar to the ones above) for rational points on the following curves:
- The ellipse \(x^2+3y^2=4\)
- The hyperbola \(x^2-2y^2=1\)
- Prove that \(x^2+y^2=15\) cannot have any rational points.
- Finish the proof that \(x^3-117y^3=5\) has no integer solutions.
- Get the tangent line to the Dudeney curve and find the point of intersection; why can it not give an answer to the original problem?
- Prove that the Bachet equation cannot have any solutions with \(x\) or \(y\) even.
- Why can the Pell equation (\(x^2-dy^2=1\)) not have any (nontrivial) solutions if \(d\) happens to be a perfect square?
- Show that the Pell equation with \(d=1\) (\(x^2-y^2=1\)) has only two solutions. Generalize this to when \(d\) happens to be a perfect square.
- Show that the equation \(x^3=y^2-999\) has no integer solutions.
- Look up the current best known bound on the number of integer points on a Mordell equation curve.
- Verify that if \[x_0^2-ny_0^2=k\text{ and }x_1^2-ny_1^2=\ell\] then \[x=x_0x_1+ny_0y_1,\; y=x_0y_1+y_0x_1\text{ solves }x^2-ny^2=k\ell\; .\]
- Explain why the previous problem reduces to the (algebraic) method we used in the examples where we were trying to use a tangent line to find more integer solutions to \(x^2-5y^2=1\).
- Find a non-trivial integer solution to \(x^2-17y^2=-1\), and use it to get a nontrivial solution to \(x^2-17y^2=1\).
- Recreate the above geometric construction using tangent lines on the hyperbola with \(x^2-5y^2=1\), and use it find three (positive) integer points on this curve with at least two digits for both \(x\) and \(y\). Yes, you will have to find a non-trivial solution on your own; it's not hard, there is one with single digits.
