MathDB

f(x) = ax^2 + bx + c is a perfect square for 2p - 1 integers

Source: IMO ShortList 1991, Problem 14 (POL 3)

August 15, 2008

modular arithmeticnumber theoryquadraticsPerfect SquareDiscriminantIMO Shortlist

Problem Statement

Let

a, b, c

be integers and

p

an odd prime number. Prove that if f(x) \equal{} ax^2 \plus{} bx \plus{} c is a perfect square for 2p \minus{} 1 consecutive integer values of

x,

then

p

divides b^2 \minus{} 4ac.