Problem 5

Consider an infinite sequence

a_1,a_2,\ldots

of positive integers such that

100!(a_m+a_{m+1}+\cdots+a_n)

is a multiple of

a_{n-m+1}a_{n+m}

for all positive integers

m,n

with

m\le n

. Prove that the sequence is either bounded or linear. A sequence is bounded if there is a constant

N

such that

a_n<N

for every positive integer

n

, and it is linear if

a_n=n\cdot a_1

for every positive integer

n

Step 3 of 10: Propagate additivity along an arithmetic progression

Previous step Next step

a_{km+n}=ka_m+a_n,\qquad a_{m+kn}=a_m+ka_n\quad(k\ge1)

Detailed analysis

Suppose $L_{m,n}>c^2(a_m+a_n)$ and write $x=a_m$ , $y=a_n$ . Step 2 gives $a_{m+n}=x+y$ . Moreover, $\operatorname{lcm}(x,x+y)=\operatorname{lcm}(x,y)(x+y)/y>c^2(2x+y)$ , and the analogous inequality holds with $x,y$ exchanged. Thus Step 2 can be applied repeatedly, yielding $a_{km+n}=ka_m+a_n$ and $a_{m+kn}=a_m+ka_n$ for every $k\ge1$ .