1

Do the algebra which we skipped in Exercise 15.7.1.

2

Do the algebra which we skipped in Example 15.1.4.

3

Find a parametrization (similar to Fact 15.1.1) for rational points on the following curves:

• The ellipse \(x^2+3y^2=4\)

• The hyperbola \(x^2-2y^2=1\)

4

Finish proving (Fact 15.1.5) that \(x^2+y^2=15\) cannot have any rational points.

5

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

6

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

7

Fill in some (or all) of the details of Theorem 15.3.3.

8

Fill in the details of divisibility to finish Euler's ‘proof’ of Fact 15.3.4.

9

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

10

Get the tangent line 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?

11

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

12

Fill in the details of Example 15.5.4, and find an integer point with even bigger values.

13

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.

14

Show that the algebra in Fact 15.6.2 yields the same formulas as Proposition 15.6.1.

15

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\; .\end{equation*}

16

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.

17

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

18

Recreate the geometric constructions in Section 15.5 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.