Seven Smurfs live in seven mushroom houses. There is a tunnel between every pair of houses, so from any house you can walk to any other house. One of the Smurfs, Clumsy, starts walking from his house, but he must not use the same tunnel more than once. He keeps walking until he reaches a house where all the tunnels have already been used. Where will Clumsy’s journey end?
A battle of the captains was held at a maths battle. The task was to write the smallest number that is divisible by \(60\) and it’s digits are only ’\(1\)’s or ’\(0\)’s. What is the answer?
Suppose that \(x_1+y_1\sqrt{d}\) and \(x_2+y_2\sqrt{d}\) give solutions to Pell’s equation \(x^2-dy^2=1\) and \(x_1,x_2,y_1,y_2\geq 0\). Show that the following are equivalent:
\(x_1+y_1\sqrt{d} < x_2+y_2\sqrt{d}\),
\(x_1<x_2\) and \(y_1<y_2\),
\(x_1<x_2\) or \(y_1<y_2\).
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 \(y_{n+2} = 2x_1y_{n+1}-y_n\).
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 \(y\).
\(x^2 - 7y^2 = 1\);
\(x^2 - 7y^2 = 29\).