the iteration map $x*\left\lceil x \right\rceil$, where the initial term is a non integer rational number $b/a$ with $b>a$ and $a>1$. the growth rate of this iteration is similar to the growth rate of Fermat numbers(!). If the initial term is (for example) $200/199$, then how we know that after $1000$ steps the number becomes an integer or not ? Because the number at $1000$ steps is very very HUGE number(!). I tried residues method, but that didn't work at all !. Is there a simple method to check this out ?