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Odd natural numbers a,b; a|b^2+2 and b|a^2+2

Source: Vietnamese National Mathematical Olympiad 2012-P6

February 8, 2012

inductionalgebrasystem of equationsnumber theory proposednumber theory

Problem Statement

Consider two odd natural numbers

a

and

b

where

a

is a divisor of

b^2+2

and

b

is a divisor of

a^2+2.

Prove that

a

and

b

are the terms of the series of natural numbers

\langle v_n\rangle

defined by

v_1 = v_2 = 1; v_n = 4v_ {n-1}-v_{n-2} \ \ \text{for} \ n\geq 3.