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Message from discussion a number theory problem
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Tim Little  
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 More options Sep 6 2012, 12:59 am
Newsgroups: sci.math, alt.math.recreational, sci.crypt
From: Tim Little <t...@little-possums.net>
Date: 06 Sep 2012 04:59:26 GMT
Local: Thurs, Sep 6 2012 12:59 am
Subject: Re: a number theory problem
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On 2012-09-06, Tim Little <t...@little-possums.net> wrote:

> It does seem likely that every integer can be generated this way,

All the positive ones, anyway.

> The number of square roots required would become insanely large very
> quickly.

Just to follow up on this, 3! = 6 (which also yields 2 and 1 with
square roots).  3!! = 720, reducing to 5 in two square root.  3!!! has
1747 digits and reduces down to 7 in 12 square roots.  The smallest
unreached number at this point is 4.

3!!!! has about 4.54*10^1749 digits.  That's still not too bad, it
requires only about 5813 square roots to get down to a single-digit
number (which is just 3 again).

The next iterate, 3!!!!! has more digits than I can conveniently
express and requires more than 10^1750 square roots to get down to any
writable integer.  Perhaps someone might be cleverly able to tell
whether it hits 4 or not in its sequence of square roots, but I
certainly can't.

It only gets worse from there.

--
Tim


 
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