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 7 of 10: A high-lcm pair forces the whole sequence to be linear

Previous step Next step

\operatorname{lcm}(a_{pd},a_{kd-1})>c^2(a_{pd}+a_{kd-1}),\qquad \gcd(pd,kd-1)=1\Longrightarrow a_j=ja_1

Detailed analysis

Choose $k$ so large that $b=a_{kd-1}>c^2t$ , then choose a prime $p>b$ with $p\nmid(kd-1)$ and sufficiently large that $pb>c^2(pt+b)$ . Since $a_{pd}=pt$ , we have $\operatorname{lcm}(a_{pd},b)\ge pb$ , so Step 2 applies. The pair $(pd,kd-1)$ has gcd $1$ , because $\gcd(d,kd-1)=1$ and $p\nmid(kd-1)$ . Steps 3–5 therefore give $a_j=j a_1$ for all $j$ (in the normalized sequence).