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 4 of 10: The two chosen terms have proportional indices

Previous step Next step

ma_n=na_m,\qquad d=\gcd(m,n),\qquad a_m=\frac md t,\quad a_n=\frac nd t

Detailed analysis

The two formulas in Step 3 evaluate the same term: $a_{n+(n+1)m}=a_{m+(m+1)n}$ . Therefore $ma_n=na_m$ . With $d=\gcd(m,n)$ , coprimality after dividing by $d$ gives an integer $t>0$ such that $a_m=(m/d)t$ and $a_n=(n/d)t$ .