Section15.8Exercises

  1. Find a parametrization (similar to the ones above) for rational points on the following curves:
    • The ellipse \(x^2+3y^2=4\)
    • The hyperbola \(x^2-2y^2=1\)
  2. Prove that \(x^2+y^2=15\) cannot have any rational points.
  3. Finish the proof that \(x^3-117y^3=5\) has no integer solutions.
  4. 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?
  5. Prove that the Bachet equation cannot have any solutions with \(x\) or \(y\) even.
  6. Why can the Pell equation (\(x^2-dy^2=1\)) not have any (nontrivial) solutions if \(d\) happens to be a perfect square?
  7. 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.
  8. Show that the equation \(x^3=y^2-999\) has no integer solutions.
  9. Look up the current best known bound on the number of integer points on a Mordell equation curve.
  10. Verify that if \[x_0^2-ny_0^2=k\text{ and }x_1^2-ny_1^2=\ell\] then \[x=x_0x_1+ny_0y_1,\; y=x_0y_1+y_0x_1\text{ solves }x^2-ny^2=k\ell\; .\]
  11. 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 \(x^2-5y^2=1\).
  12. Find a non-trivial integer solution to \(x^2-17y^2=-1\), and use it to get a nontrivial solution to \(x^2-17y^2=1\).
  13. 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.