MathDB

Two different integers

u

and

v

are written on a board. We perform a sequence of steps. At each step we do one of the following two operations:
(i) If

a

and

b

are different integers on the board, then we can write

a + b

on the board, if it is not already there. (ii) If

a

b

and

c

are three different integers on the board, and if an integer

x

satisfies

ax^2 +bx+c = 0

, then we can write

x

on the board, if it is not already there.
Determine all pairs of starting numbers

(u, v)

from which any integer can eventually be written on the board after a finite sequence of steps.

Problem Statement