Problems

Prove that the only solution to \(5^a-3^b=2\) with \(a,b\) being positive integers is \(a=b=1\).

Show that Pell’s equation \(x^2-dy^2=1\) has a nontrivial solution.

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 \(x=2m+1\), \(y=2n\).

A rectangle has a perimeter of \(1\). Is it possible that its area is larger than \(1000\)?

Zahra has a \(3\times 3\) grid of little squares. Can she write the numbers \(2,4,6,7,8,10,12,14,16\) inside the little squares - using each number exactly once - so that the sum of the three numbers in every row is the same?

Sam was playing on the \(3\times 3\) lights out board, and starting from all lights off, he pressed every single button on the board. What light pattern did he get in the end?

Which of the triangles has the smallest area?

Find a quiet plan for the \(1\times 5\) “Lights Out" board.

There are \(57\) buttons placed evenly around a circle, each with a light underneath it. Pressing a button changes the state of its own light and also the lights under its two neighbouring buttons. Changing state means that a light which is on becomes off, and a light which is off becomes on. If all the lights start off, what is the smallest number of presses that need to be done so that the entire circle lights up?