Problem 2

Find all integers

n \ge 3

for which there exist real numbers

a_1, a_2, \ldots, a_n

satisfying

a_{n+1}=a_1

a_{n+2}=a_2

, and

a_ia_{i+1}+1=a_{i+2}

for

i=1,2,\ldots,n

Step 5 of 7: If

3\nmid n

the sequence is constant

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\gcd(n,3)=1\ \Longrightarrow\ a_1=a_2=\cdots=a_n

Detailed analysis

The relation $a_i=a_{i+3}$ for every $i$ means $a_i$ depends only on the residue of $i$ modulo $\gcd(n,3)$ , because repeatedly shifting an index by $3$ modulo $n$ cycles exactly through its residue class modulo $\gcd(n,3)$ before returning to itself. Since $\gcd(n,3)\in\{1,3\}$ , the case $3\nmid n$ forces $\gcd(n,3)=1$ , so every index lies in one residue class and $a_1=a_2=\cdots=a_n$ .