Pen problem

if you are refering to the problem in PEN below, it is not the same PEN N17 wrote: Suppose that $a$ and $b$ are distinct real numbers such that \[a-b, \; a^{2}-b^{2}, \; \cdots, \; a^{k}-b^{k}, \; \cdots\]are all integers. Show that $a$ and $b$ are integers. solved here

parmenides51 wrote: Let $u$ and $v$ be positive rational numbers with $u \ne v$. Assume that there are infinitely many positive integers $n$ with the property that $u^n - v^n$ are integers. Prove that $u$ and $v$ are integers. Let $u=\frac{x}{z}$ and $v=\frac{y}{z}$, where $\gcd (x,y,z)=1$ then $z^n\mid x^n-y^n$ for infinitely $n$. Assume that there exists a prime $p$ divides $z$. Let $h$ be the order of $x/y$ modulo $p$. By the LTE lemma, $$n\leq \nu_p (x^h-y^h)+\nu_p(n)-\nu_p(h),$$absurb as $\lim (n-\nu_p (n))=\infty$, similarly to $p=2$. Hence $z=1$ and the conclusion follows.