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From: dole@spar.UUCP (Harry Dole)
Newsgroups: net.math
Subject: Re: Can you prove/disprove this?
Message-ID: <687@spar.UUCP>
Date: Mon, 2-Dec-85 14:53:51 EST
Article-I.D.: spar.687
Posted: Mon Dec  2 14:53:51 1985
Date-Received: Thu, 5-Dec-85 05:39:47 EST
References: <34080@lanl.ARPA>
Reply-To: dole@max.UUCP (Harry Dole)
Distribution: net
Organization: Yoyodyne Semicunduktur - Where the Futur is Tomorro
Lines: 27
Summary: 

In article <34080@lanl.ARPA> dxm@lanl.ARPA writes:
>that numbers of the form 
>
>		x = 2^(n-1) * (2^n - 1)
>
>are perfect if n is a prime number.  This was true up to the largest
>integer the machine could represent.  
>
>
> Doug Miller                     dxm@lanl.arpa
>                                 ....!ihnp4!lanl!dxm

This looks like the standard characterization of perfect numbers but
not quite: 

	If 2^n - 1 is prime, then x = ( 2^n - 1 ) * ( 2^(n-1) )
	is perfect and every even perfect number has this form.

It is rumoured that even Euclid knew the first part of this one.  Note
that for n=4 we have x = 120.  Let sigma(y) denote the sum of all
positive divisors of y - e.g., sigma(6)=1+2+3+6.  Then y is perfect
if sigma(y)=2*y.  However, sigma(120)=3*120 so 120 is referred to as
3-perfect, not perfect.  Did you run this on a 2 bit computer?

				Harry Dole

	- Long Live the Categorical Imperative