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 4 of 8: If a valuation reaches

v_p(a_1)

, it descends

Previous step Next step

v_p(a_k)\ge c=v_p(a_1)\Longrightarrow c\le v_p(a_{n+1})\le v_p(a_n)\quad(n\ge k)

Detailed analysis

Now suppose $p\mid a_1$ and put $c=v_p(a_1)>0$ . If $v_p(a_k)\ge c$ for some $k>N$ , induction gives $v_p(a_n)\ge c$ and $v_p(a_{n+1})\le v_p(a_n)$ thereafter. Indeed, if the next valuation were below $c$ , the first and third terms of $T_n$ would have unequal valuations (the possible equality would force $v_p(a_n)<c$ ), leaving a unique negative valuation; and if it exceeded the current one, the third term would be uniquely negative. Thus the integer sequence of valuations is eventually constant.