Pell's equation (Table of contents)

Problems

Problem 1

Solve in integers the equation \(x^2+y^2-1=4xy\).

Problem 2

For a given integer \(d\), solve \(x^2-dy^2=1\) in the set of rational numbers.

Problem 3

Let \((x,y)=(a,b)\), \(a,b\in\mathbb{N}\) be the smallest integer solution of \(x^2-dy^2=1\). Consider the sequence defined by \(y_0=0\), \(y_1=b\), \(y_{n+1}=2ay_n-y_{n-1}\) for \(n\geq1\). Show that \(ay_n^2+1\) is a square for each \(n\). Show that if \(ay^2+1\) is a square for some \(y\in\mathbb{N}\), then \(y=y_n\) for some \(n\).

Problem 4

Prove that \(5x^2+4\) or \(5x^2-4\) is a perfect square if and only if \(x\) is a term in the Fibonacci sequence.

Problem 5

Find all \(n\in\mathbb{N}\) such that \(\displaystyle\binom n{k-1}= 2\binom nk+\binom n{k+1}\) for some natural number \(k < n\).

Problem 6

Let \(a\in\mathbb{N}\) and \(d=a^2-1\). If \(x,y\) are integers and the absolute value of \(m=x^2-dy^2\) is less than \(2a+2\), prove that \(|m|\) is a perfect square.

Problem 7

Prove that if \(m=2+2\sqrt{28n^2+1}\) is an integer for some \(n\in\mathbb{N}\), then \(m\) is a perfect square.

Problem 8

Prove that if the difference of two consecutive cubes is \(n^2\), \(n\in\mathbb{N}\), then \(2n-1\) is a square.

Problem 9

If \(n\) is an integer such that \(3n+1\) and \(4n+1\) are both squares, prove that \(n\) is a multiple of 56.

Problem 10

Prove that the equation \(x^2-dy^2=-1\) is solvable in integers if and only if so is \(x^2-dy^2=-4\).

Problem 11

Let \(p\) be a prime. Prove that the equation \(x^2-py^2=-1\) has integral solutions if and only if \(p=2\) or \(p\equiv 1\) (mod 4).

Problem 12

If \(p\) is a prime of the form \(4k+3\), show that exactly one of the equations \(x^2-py^2=\pm2\) has an integral solution.

Problem 13

Prove that \(3^n-2\) is a square only for \(n=1\) and \(n=3\).

Problem 14

Prove that if \(\displaystyle \frac{x^2+1}{y^2}+4\) is a perfect square, then this square equals 9.