MathDB

Sequence NT

Source: Vietnam TST 2023 P4

April 14, 2023

number theory

Problem Statement

Given are two coprime positive integers

a, b

with

b

odd and

a>2

. The sequence

(x_n)

is defined by

x_0=2, x_1=a

and

x_{n+2}=ax_{n+1}+bx_n

for

n \geq 1

. Prove that:

a)

If

a

is even then there do not exist positive integers

m, n, p

such that

a

is a positive integer.

b)

If

a

is odd then there do not exist positive integers

m, n, p

such that

mnp

is even and

\frac{x_m} {x_nx_p}

is a perfect square.

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