MathDB

There exist an infinite number of pairs of integers (x, y)

Source: IMO Shortlist 1992, Problem 1

August 13, 2008

algebrapolynomialVietaquadraticsnumber theoryIMO Shortlist

Problem Statement

Prove that for any positive integer

m

there exist an infinite number of pairs of integers

(x, y)

such that (i)

x

and

y

are relatively prime; (ii)

y

divides x^2 \plus{} m; (iii)

x

divides y^2 \plus{} m. (iv) x \plus{} y \leq m \plus{} 1\minus{} (optional condition)