gobathegreat wrote: Prove that for every pair of positive integers $(m,n)$, bigger than $2$, there exists positive integer $k$ and numbers $a_0,a_1,...,a_k$, which are bigger than $2$, such that $a_0=m$, $a_1=n$ and for all $i=0,1,...,k-1$ holds $$ a_i+a_{i+1} \mid a_ia_{i+1}+1$$ It's wrong of course! Take $m=n=3$. Then we need $k \ge 1$ and hence for $i=0$ we have $a_0+a_1 \mid a_0a_1+1$ i.e. $6 \mid 10$ which certainly isn't true.

from one solution I saw, his meaning is $a_k=n$, which directed a wonderful result. hint: link m and m+1