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Exercises 15.7 Exercises

Find a parametrization (similar to Fact 15.1.2) for rational points on the following curves.

3.

The ellipse \(x^2+3y^2=4\text{.}\)

4.

The hyperbola \(x^2-2y^2=1\text{.}\)

5.

Finish proving (Fact 15.1.8) that \(x^2+y^2=15\) cannot have any rational points.

6.

Finish the proof that \(x^3-117y^3=5\) has no integer solutions, looking modulo nine.

7.

Show that the equation \(x^3=y^2-999\) has no integer solutions. (This is also Exercise 7.7.14.)

9.

Use Theorem 15.3.4 to come up with three Mordell curves we haven't yet mentioned which have no integer solutions.

10.

Fill in the details of divisibility to finish Euler's ‘proof’ of Fact 15.3.5.

11.

Look up the current best known bound on the number of integer points on a Mordell equation curve.

12.

Get the tangent line at \((2,1)\) to the Dudeney curve (see Question 15.2.1) and find the point of intersection; why can it not give an answer to the original problem?

13.

Research Boyer's or Stigler's laws. What is the most egregious example of this, in your opinion?

14.

Fill in the details of Example 15.5.8, and then find an integer point with even bigger values than in that example.

15.

Show that the Pell equation with \(d=1\) (\(x^2-y^2=1\)) has only two integer solutions. Generalize this to when \(d\) happens to be a perfect square.

16.

Show that algebraically expanding the identity in Fact 15.6.2 to solve for \(x_1,y_1\) yields the formulas for \(x\) and \(y\) in the proof of Proposition 15.6.1.

17.

Verify that if

\begin{equation*} x_0^2-ny_0^2=k\text{ and }x_1^2-ny_1^2=\ell \end{equation*}

then

\begin{equation*} x=x_0x_1+ny_0y_1,\; y=x_0y_1+y_0x_1\text{ solves }x^2-ny^2=k\ell\text{.} \end{equation*}
18.

Explain why the previous problem reduces to the method from Section 15.5 where we were trying to use a tangent line to find more integer solutions.

19.

Find a non-trivial integer solution to \(x^2-17y^2=-1\text{,}\) and use it to get a nontrivial solution to \(x^2-17y^2=1\text{.}\)

20.

Recreate the geometric constructions in Section 15.5 using tangent lines on the hyperbola with \(x^2-5y^2=1\text{,}\) and use it find three (positive) integer points on this curve with at least two digits for both \(x\) and \(y\text{.}\) Yes, you will have to find a non-trivial solution on your own; it's not hard, there is one with single digits.

21.

Recall Remark 14.1.9 that the set of primitive Pythagorean triples can form a group, which evidently might be related to the graphs of circles \(x^2+y^2=c^2\text{.}\) Find the article [E.7.38] connecting the same set, as a group under a different multiplication, to the hyperbolas \(x^2-y^2=a^2\text{,}\) and compare this to the story in Section 15.6. Which ones seems more interesting, or more computable?