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 2 of 10: A large least common multiple forces additivity

Previous step Next step

L=\operatorname{lcm}(a_r,a_s,a_{r+s})\mid c(a_{r+s}-a_r-a_s),\qquad L_{r,s}:=\operatorname{lcm}(a_r,a_s)>c^2(a_r+a_s)\Rightarrow a_{r+s}=a_r+a_s

Detailed analysis

The relations in Step 1 show that $c(a_{r+s}-a_r-a_s)$ is divisible by each of $a_r$ , $a_s$ , and $a_{r+s}$ , hence by $L=\operatorname{lcm}(a_r,a_s,a_{r+s})$ . Also $a_{r+s}\le c(a_r+a_s)$ . If $L_{r,s}>c^2(a_r+a_s)$ , then the two positive congruent numbers $ca_{r+s}$ and $c(a_r+a_s)$ are both smaller than $L_{r,s}$ , so they are equal, proving $a_{r+s}=a_r+a_s$ .