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 3 of 8: Primes absent from

a_1

decrease

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p\nmid a_1\Longrightarrow v_p(a_{n+1})\le v_p(a_n)

Detailed analysis

If $p\nmid a_1$ , then $v(\frac{a_{n+1}}{a_1})\ge0$ and $v(\frac{a_n}{a_1})\ge0$ . Since $T_n$ is an integer, the last term must also have nonnegative valuation: $v(\frac{a_n}{a_{n+1}})\ge0$ . Hence $v_p(a_{n+1})\le v_p(a_n)$ for every $n>N$ .