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.


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