paladin8 wrote: Call two integers $ a$ and $ b$ "linked" if there exists a sequence as described in the question. We wish to show that all integers $ a,b > 2$ are linked. If we can show that $ m$ and $ m + 1$ are linked for any integer $ m > 2$ in a finite number of terms (links), the problem is solved. Consider the following sequence: $ m$, $ m(m - 1)$, $ m(m - 1)(m - 2)$, $ 2m(m - 1)$, $ 2m(m + 1)$, $ m(m + 1)(m - 1)$, $ m(m + 1)$, $ m + 1$, which can easily be shown to satisfy the property in the question. Thus $ m$ and $ m + 1$ are always linked for $ m > 2$. Hence, by induction, any integers $ a, b > 2$ are linked, as desired.

A similar approach by my friend: Case 1:$ a$ is odd $ a\rightarrow a(a-1)\rightarrow a(a+1)\rightarrow a+1$ Case 2:$ a$ is even $ a \rightarrow a(\frac{a}{2}-1) \rightarrow a(\frac{a}{2}+1)\rightarrow a+2 \rightarrow (a+2)(a+1) \rightarrow a(a+1) \rightarrow a+1$ QED