text: # Number theory, prime number
  
  • Sep 17th 2006, 04:58 PM
  suedenation
  Number theory, prime number
  Show that if p is a prime in the arithmetic progression 3n + 1, n=1, 2, 3,..., then it is also in the arithmetic progression 6n + 1, n=1, 2, 3,......
  
  Thanks very much for your help!!!:)
  • Sep 17th 2006, 08:11 PM
  CaptainBlack
  Quote:
  
  Originally Posted by suedenation
  Show that if p is a prime in the arithmetic progression 3n + 1, n=1, 2, 3,..., then it is also in the arithmetic progression 6n + 1, n=1, 2, 3,......
  
  Thanks very much for your help!!!:)
  
  Suppose p is a prime in the progression 3n+1, n=1, 2, 3... Then there
  exists a positive integer k such that:
  
  p=3k+1.
  
  Now if p>=2, then p is odd as the only even prime is 2. Therefore p-1 is
  even, so 3k is even, so k is even, so there exists an integer m>0 such that:
  
  p-1=3.2.m=6m,
  
  so for some m>0, p=6m+1, therefore p is in the progression 6n+1, n=1, 2, ..
  
  RonL
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