MathDB

Infinitely many n exist such that b divides a_n

Source: Balkan MO ShortList 2008 N5

April 5, 2020

Problem Statement

Let

(a_n)

be a sequence with

a_1=0

and

a_{n+1}=2+a_n

for odd

n

and

a_{n+1}=2a_n

for even

n

. Prove that for each prime

p>3

, the number \begin{align*} b=\frac{2^{2p}-1}{3} \mid a_n \end{align*} for infinitely many values of

n