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 5 of 8: If it never reaches

c

, it ascends

Previous step Next step

v_p(a_n)<c\ \forall n>N\Longrightarrow v_p(a_{n+1})\ge v_p(a_n)

Detailed analysis

In the remaining case, $x=v_p(a_n)<c$ and $y=v_p(a_{n+1})<c$ . The first two terms of $T_n$ have valuations $y-c$ and $x-c$ , both negative. For their sum with the third term to be integral, the minimum valuation must occur twice. Equality of the first two gives $x=y$ ; equality of the first and third gives $2y=x+c$ , hence $y>x$ ; equality of the second and third would give $y=c$ , impossible. Thus $y\ge x$ , and the valuations are bounded above by $c$ , so they stabilize.