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From: johns@phred.UUCP
Newsgroups: sci.crypt
Subject: Spacing of Prime Numbers
Message-ID: <1392@phred.UUCP>
Date: Tue, 28-Apr-87 15:48:56 EDT
Article-I.D.: phred.1392
Posted: Tue Apr 28 15:48:56 1987
Date-Received: Fri, 1-May-87 01:00:01 EDT
Organization: Physio Control Corp., Seattle WA
Lines: 39
Keywords: Primes


 This may not be news to readers  of this group, but I thought the follow-
 ing tidbit was interesting. 
  
          The May issue of DISCOVER  magazine  presented a cute problem as
 part of its brain teaser section  on  the  last page of the magazine. The
 question was (I'm paraphrasing it):
  
          Prove there is a  sequence  of at least one million
          consecutive integers, none of whom are prime.
          
 The answer is surprisingly easy:
  
 ----------------------(Spoiler Follows)----------------------------

 Let M = (3 * 4 * 5 * ... * 1,000,003) = (1,000,003!) / 2
  
 Then the consecutive sequence of one million integers consisting of:
  
          M3 = (M + 3)   (that's "M sub three", not "M times three")
          M4 = (M + 4)
          M5 = (M + 5)
  
          ... etc, etc up to  M1000003 = (M + 1,000,003)
  
 contains no primes because 3 is a factor of M3, 4 is a factor of M4, and
 so on, and so on... until 1000003 is a factor of M1000003.
  
 I had always guessed primes were  more  closely spaced than that. 
 (But then, number theory is not my forte.)
  
 BTW, DISCOVER is  a  great  magazine  for explaining technical
 subjects using language almost anyone  can understand. It's usually good
 reading for subjects outside  one's  specialty.  (Such as number theory,
 for me).
  
 John Stice.............
 
 usual disclaimers apply..... etc.