Problems

If Pell’s equation \(x^2-dy^2 = 1\) has a nontrivial solution \((x_1,y_1)\), show that it has infinitely many distinct solutions.

Show that there are infinitely many triples of consecutive integers, each of which is a sum of the square of two integers.

Suppose that Pell’s equation \(x^2-dy^2=1\) has a solution \((x_1,y_1)\) where \(x_1,y_1\) are positive and \(y_1\) is minimal among all solutions with positive \(x,y\). Show that if \(x+y\sqrt{d}\) gives a solution to \(x^2-dy^2=1\), then \(x+y\sqrt{d}=\pm(x_1+y_1\sqrt{d})^k\) for some integer \(k\).

Suppose that \(x_1+y_1\sqrt{d}\) gives a solution to Pell’s equation \(x^2-dy^2=1\). Define a sequence \(x_n+y_n\sqrt{d} = (x_1+y_1\sqrt{d})^n\). Show that we have the recurrence relations \(x_{n+2} = 2x_1x_{n+1}-x_n\) and \(y_{n+2} = 2x_1y_{n+1}-y_n\).

For the following equations, find the integer solution \((x,y)\) with the smallest possible absolute value of \(y\).

• \(x^2 - 7y^2 = 1\);

• \(x^2 - 7y^2 = 29\).

Find the integer solution \((x,y)\) with the smallest possible absolute value of \(y\). \(x^2 - 2y^2 = 1\);

This equation helps to find all the square-triangular numbers, namely all the numbers that are perfect squares and can be represented as the sum \(1+2+3+...m\) for some \(m\). Finding such a number is equivalent to finding a solution to the equation: \(2n^2 = m(m+1)\). Or finding a solution to the Pell’s equation \(x^2-2y^2 = 1\) for \(x=2m+1\), \(y=2n\).

Which of the triangles has the smallest area?

Without using any wolves, show that Robinson’s goat can only graze shapes that are convex (that means, whenever you pick two points inside the shape, the whole line between them also lies inside). But if Robinson is allowed to use as many wolves as he likes, this restriction disappears. Show that in this case, he can make the goat graze in the shape of any polygon at all.

In a field, there are two pegs, \(A\) and \(B\), placed \(15\) metres apart. Each peg has a small ring on top, and a rope can slide freely through these rings.
You have one rope and two goats that want to graze the grass, but they will fight each other if they can both reach the same spot.
The rope may be arranged in any way you choose: it could pass through both rings, only one, or neither, depending on your setup.
For what lengths of rope can you arrange things so that the goats cannot fight each other?

Long before meeting Snow White, the seven dwarves lived in seven different mines. There is an underground tunnel connecting any two mines. All tunnels were separate, so you could not start in one tunnel and somehow end up in another. Is it possible to walk through every tunnel exactly once without retracing your path?