MathDB

Gcd of some consecutive terms of a recursive sequence

Source: Swiss MO 2023/3

March 12, 2023

number theorygreatest common divisor

Problem Statement

Let

x,y

and

a_0, a_1, a_2, \cdots

be integers satisfying

a_0 = a_1 = 0

, and

a_{n+2} = xa_{n+1}+ya_n+1

for all integers

n \geq 0

. Let

p

be any prime number. Show that

\gcd(a_p,a_{p+1})

is either equal to

1

or greater than

\sqrt{p}