From: landsbur@troi.cc.rochester.edu (Steven E. Landsburg)
Subject: Re: A Brain Teaser
Date: 30 Jun 1999 15:48:42 GMT
Newsgroups: sci.math
Keywords: [missing]

In response to a query about expressing every natural number with
only three 2's, seraphsama@aol.com writes:

>n = log_2{log_[sqrt(sqrt(..sqrt(2))..)] 2}, as many sqrt's necessary


This leaves open the question of whethere one can write every natural
number with only *one* 2, together with the symbols !, sqrt, and []
(greatest integer).  

Mel Hochster once raised this question:  Given a natural number n,
do there exist natural numbers p and q such that

 n = [sqrt(sqrt(sqrt(...2!!!...!)))]

with p sqrts and q !'s. 

Steven E. Landsburg
steven@landsburg.com
==============================================================================

From: landsbur@troi.cc.rochester.edu (Steven E. Landsburg)
Subject: Re: A Brain Teaser
Date: 30 Jun 1999 19:41:24 GMT
Newsgroups: sci.math

Eep! I asked:

>>Mel Hochster once raised this question:  Given a natural number n,
>>do there exist natural numbers p and q such that
>>
>> n = [sqrt(sqrt(sqrt(...2!!!...!)))]
>>
>>with p sqrts and q !'s. 
>>

and Virgil answered:

>Since 2! = 2, the factorials can be ignored, and since the iterated square
>roots of 2 are still strictly between 1 and 2, the trivial answer is no.
>
>-- 

Clearly, I should have used a 3, not a 2.  In fact, Hochster
originally asked the question for 4, motivated by the old riddle
about representing every natural number with just four 4's.

Steven E. Landsburg
steven@landsburg.com