Problem 5

Let

a_1,a_2,\ldots

be an infinite sequence of positive integers. Suppose there is an integer

N>1

such that, for every

n\ge N

, the number

\frac{a_1}{a_2}+\frac{a_2}{a_3}+\cdots+\frac{a_{n-1}}{a_n}+\frac{a_n}{a_1}

is an integer. Prove that there is a positive integer

M

such that

a_m=a_{m+1}

for all

m\ge M

Step 7 of 8: The tail is a divisibility chain

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a_{n+1}\mid a_n\quad\text{for all sufficiently large }n

Detailed analysis

For primes $p\nmid a_1$ , Step 3 gives $v_p(a_{n+1})\le v_p(a_n)$ . For primes $p\mid a_1$ , Step 6 makes the valuations equal after a common point. Therefore, for all sufficiently large $n$ , every prime exponent in $a_{n+1}$ is at most its exponent in $a_n$ , so $a_{n+1}\mid a_n$ .