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 ?

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    $\begingroup$ I am not sure I understand, so again, this site is about specific problems with Wolfram Mathematica. Is your question about problem in implementation of an algorithm in Wolfram Mathematica? $\endgroup$ – Kuba Nov 15 '18 at 10:08
  • $\begingroup$ Yes sth like that.. $\endgroup$ – Brandon J. Nov 15 '18 at 10:11
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    $\begingroup$ Please add code for what you've already tried (residues method?) to your question. $\endgroup$ – Carl Lange Nov 15 '18 at 10:27
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    $\begingroup$ @Brandon, Please use the edit option to place the iterate... example in the actual post. As it is, I almost missed seeing the one in the comments, and nearly voted to close based on lack of any concrete example. $\endgroup$ – Daniel Lichtblau Nov 15 '18 at 16:34
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    $\begingroup$ This link maybe the solution (at page $20$) $$\href{neilsloane.com/doc/apsq.pdf}$$ $\endgroup$ – WhatEVer Nov 16 '18 at 3:15

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