MathDB

For a number consists of

0

and

1

, one can perform the following operation: change all

1

into

100

, all

0

into

1

. For all nonnegative integer

n

, let

A_n

be the number obtained by performing the operation

n

times on

1

(starts with

100,10011,10011100100,\dots

), and

a_n

be the

n

-th digit(from the left side) of

A_n

. Prove or disprove that there exists a positive integer

m

satisfies the following:
For every positive integer

l

, there exists a positive integer

k\le m

satisfying

a_{l+k+1}=a_1,\ a_{l+k+2}=a_2,\ \dots,\ a_{l+k+2017}=a_{2017}

Problem Statement