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 8 of 10: Assume every least common multiple is small

Previous step Next step

\operatorname{lcm}(a_r,a_s)\le 2c^2\max(a_r,a_s)\quad\text{for all }r,s

Detailed analysis

It remains to consider the case $\operatorname{lcm}(a_r,a_s)\le c^2(a_r+a_s)$ for every pair. If $a_r\le a_s$ , write $\operatorname{lcm}(a_r,a_s)=Ma_s$ . Then $M\le2c^2$ ; set $K=2c^2$ . Thus for every $r$ and every term $s$ with value $a_s\ge a_r$ , the quotient $a_r/\gcd(a_r,a_s)$ is at most $K$ .