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 2 of 8: Fix a prime and use valuations

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v_p(x)=\text{the exponent of }p\text{ in }x

Detailed analysis

Fix a prime $p$ and write $v=v_p$ . An integer has nonnegative $v$ -valuation, and a sum of rationals with one uniquely negative valuation cannot be an integer: its valuation is then negative.