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Consecutive Non-Prime Integers


Date: 2/17/96 at 12:42:32
From: Anonymous
Subject: help in abstract algebra

For any integer n>1 there exist n consecutive non prime 
integers. I have taken n=4 and then use 24,25,26,27 as a 
specific example. However, for n large I don't see how this 
can hold.  Can you please send me some insight on the general 
case?


Date: 3/24/96 at 12:12:30
From: Doctor Steven
Subject: Re: help in abstract algebra

This is a tough problem but one that has a relatively simple 
solution once you know it (until then it'll keep you stumped 
for days).

  Take a number n>1.  Then looking at a_k = (n+1)! + k + 1, 
where k goes from 1 to n, you get:

      a_1 = (n+1)! + 2,
      a_2 = (n+1)! + 3,
      :
      :
      a_n = (n+1)! + n + 1.

  Notice we have n consecutive integers. Notice 2, 3, ...., n+1 
all divide (n+1)!.  Notice also that 2, 3, (n+1) divide 2, 3, ... 
(n+1) respectively.  So we have n consecutive composite 
integers.

-Doctor Steven,  The Math Forum

    
Associated Topics:
High School Number Theory

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