TITLE: Let $a,b$ be positive integers such that $a\mid b^2 , b^2\mid a^3 , a^3\mid b^4 \ldots$ so on , then $a=b$?
QUESTION [5 upvotes]: Let $a,b$ be positive integers such that $a\mid b^2 , b^2\mid a^3 , a^3\mid b^4 \ldots$ that is $a^{2n-1}\mid b^{2n} ; b^{2n}\mid a^{2n+1} , \forall n \in \mathbb Z^+$ , then is it true that $a=b$ ?

REPLY [3 votes]: This is a twist of another answer here in order to avoid prime factorization.
We may assume $a,b>1$. Then
$$\frac{2n-1}{2n} \leq \frac{\log b}{\log a} \leq \frac{2n+1}{2n}$$
for all $n$. Taking $n \to \infty$, we get $a=b$.