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 6 of 8: Every prime divisor of

a_1

stabilizes

Previous step Next step

p\mid a_1\Longrightarrow v_p(a_n)\text{ is eventually constant}

Detailed analysis

The two preceding cases cover every prime $p\mid a_1$ . Hence for each such prime, $v_p(a_n)$ is eventually constant. There are only finitely many prime divisors of $a_1$ , so one common index makes all these valuations constant.