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 5 of 10: Determine every term whose index is a multiple of d

Previous step Next step

a_{kd}=kt\quad(k\ge1)

Detailed analysis

Choose arbitrarily large positive $R,S$ with $Rm-Sn=kd$ , and set $x=n+(R+1)m$ , $y=m+(S+1)n$ . Then $x-y=kd$ , while Step 3 gives $a_x=(t/d)x$ and $a_y=(t/d)y$ . Since $a_x=a_{y+kd}$ divides $c(a_y+a_{kd})$ , it also divides $c(a_{kd}-tk)$ . Letting $R,S$ grow makes $a_x$ arbitrarily large, so $a_{kd}=tk$ .