MathDB

A sequence

\{a_n\}

satisfies

a_1

is a positive integer and

a_{n+1}

is the largest odd integer that divides

2^n-1+a_n

for all

n\geqslant 1

. Given a positive integer

r

which is greater than

1

. Is it possible that there exists infinitely many pairs of ordered positive integers

(m,n)

for which

m>n

and

a_m = ra_n

?
In other words, if you successfully find an

a_1

that yields infinitely many pairs of

(m,n)

which work fine, you win and the answer is YES. Otherwise you have to proof NO for every possible

a_1

.
@below, XMO stands for Xueersi Mathematical Olympiad, where Xueersi (学而思) is a famous tutoring camp in China.

Problem Statement