MathDB

trinomial has 2 real roots such |x_2-x_1|><=1/n <=> n has 2 prime divisors

Source: Germany Federal - Bundeswettbewerb Mathematik 2009, round 2, p2

April 9, 2020

number theoryquadratic trinomialtrinomialprime divisorsprimeDivisors

Problem Statement

Let

n

be an integer that is greater than

1

. Prove that the following two statements are equivalent: (A) There are positive integers

a, b

and

c

that are not greater than

n

and for which that polynomial

ax^2 + bx + c

has two different real roots

x_1

and

x_2

with

| x_2- x_1 | \le \frac{1}{n}

(B) The number

n

has at least two different prime divisors.