# Question about approximate squaring iteration [closed]

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 ?

• 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? – Kuba Nov 15 '18 at 10:08
• Yes sth like that.. – Brandon J. Nov 15 '18 at 10:11
• @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. – Daniel Lichtblau Nov 15 '18 at 16:34
• This link maybe the solution (at page $20$) $$\href{neilsloane.com/doc/apsq.pdf}$$ – WhatEVer Nov 16 '18 at 3:15