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Math Help - Number theory, prime number

  1. #1
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    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!!!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by suedenation View Post
    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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