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Integers expressible as product of elements of an interesting set

Source: The South African Mathematical Olympiad P4

July 30, 2023

number theory

Problem Statement

Let

A

be a set of real numbers satisfying the following: (a)

\sqrt(n^2+1) \in A

for all positive integers

n

, (b) if

x \in A

and

y \in A

, then

x-y \in A

. Prove that every integer can be written as a product of two different elements in

A