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From: billc@prism.UUCP
Newsgroups: sci.crypt
Subject: Re: Spacing of Prime Numbers
Message-ID: <201000001@prism>
Date: Mon, 1-Jun-87 13:46:00 EDT
Article-I.D.: prism.201000001
Posted: Mon Jun  1 13:46:00 1987
Date-Received: Tue, 9-Jun-87 07:09:04 EDT
References: <28047@rochester.ARPA>
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Nf-From: prism.UUCP!billc    Jun  1 13:46:00 1987


==/* Written  2:39 am  May 24, 1987 by falk@sun.UUCP in prism:sci.crypt */
==In article <1938@husc6.UUCP>, greg@endor.harvard.edu (Greg) writes:
==> >           Prove there is a  sequence  of at least one million
==> >           consecutive integers, none of whom are prime.
==>
==> 1000001! + n is divisible by n for 2<=n<=1000001.
==
==I'm confused.  How does this prove that there's a million consecutive
==composite numbers?  All I see is that there's a composite number with
==a million consecutive factors.
==
==-- 
==		-ed falk, sun microsystems, falk@sun.com
==terrorist, cryptography, DES, drugs, cipher, secret, decode, NSA, CIA, NRO.

	1000001! is indeed a number with a million consecutive factors.
	So how does this help us, you ask?  The point is that for any
	number between one and one million, call it n, 1000001!
	can be written as c*n, where c is 1000001!/n.  Thus, 1000001! + n
	can be written as (c+1) * n.  That means that 1000001! + n is
	not a prime.  Now that's for any of a million consecutive n's!
	Thus we've created a string of one million composites.

	This method can be used to prove that for any number n,
	a string of consecutive composites can be found.  (This
	is the principle of mathematical induction.  See a good
	reference like Calculus Vol. I by Tom Apostol.)
----
Bill Callahan	billc@mirror.TMC.COM
		{mit-eddie, ihnp4, wjh12, cca, cbosgd, seismo}!mirror!billc

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