2004 Indonesia MO Problems/Problem 6

Problem

A quadratic equation $x^2 + ax + b + 1 = 0$ with integers $a$ and $b$ has roots which are positive integers. Prove that $a^2 + b^2$ is not a prime.

Solution

Let $r$ and $s$ be the two positive roots of the quadratic. By Vieta's Formulas, $r + s = -a$ $rs-1 = b$ Square both equations to get $r^2 + 2rs + s^2 = a^2$ $r^2s^2 - 2rs + 1 = b^2$ Add the two equations to get $a^2 + b^2 = r^2 + 2rs + s^2 + r^2s^2 - 2rs + 1$ $a^2 + b^2 = r^2 + s^2 + r^2s^2 + 1$ The right hand side can be factored into $(r^2 + 1)(s^2 + 1)$ . Since $r,s > 0$ , $r^2 + 1 > 1$ and $s^2 + 1 > 1$ , so $a^2 + b^2$ can never be a prime since it can be written as the product of two integers greater than one.

2004 Indonesia MO (Problems)
Preceded by Problem 5	1 • 2 • 3 • 4 • 5 • 6 • 7 • 8	Followed by Problem 7
All Indonesia MO Problems and Solutions

Page

Toolbox

Search

2004 Indonesia MO Problems/Problem 6

Problem

Solution

See Also