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Math Help - Prime Numbers

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    Prime Numbers

    Prove that if p, a prime number, is in the arithmetic progression of 3n + 1, n = 1, 2, 3, ..., then its also in the arithmetic progression of 6n + 1, n = 1, 2, 3, ....
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    Quote Originally Posted by Ideasman View Post
    Prove that if p, a prime number, is in the arithmetic progression of 3n + 1, n = 1, 2, 3, ..., then its also in the arithmetic progression of 6n + 1, n = 1, 2, 3, ....
    It doesn't matter whether the number is prime or not. The entire series 6n + 1 is contained within 3n + 1. To see this suppose we have a number k = 6n + 1. What is n' for k = 3n' + 1?

    k = 6n + 1 = 3n' + 1

    6n + 1 = 3n' + 1

    6n = 3n'

    n' = 2n.

    Since there is an n' for every n, thus every number 6n + 1 can be written in the form 3n' + 1.

    -Dan
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    Quote Originally Posted by topsquark View Post
    It doesn't matter whether the number is prime or not. The entire series 6n + 1 is contained within 3n + 1. To see this suppose we have a number k = 6n + 1. What is n' for k = 3n' + 1?

    k = 6n + 1 = 3n' + 1

    6n + 1 = 3n' + 1

    6n = 3n'

    n' = 2n.

    Since there is an n' for every n, thus every number 6n + 1 can be written in the form 3n' + 1.

    -Dan
    You have to be careful, Dan.

    You have:

    {6n + 1 : n is a pos integer} "is a subset of (although not equal) " {3n + 1 : n is a pos int}

    Yes I will agree with that;

    Looking at the 3n + 1 set;

    Given a prime in the largest set, we want to know if it a prime in the smaller set, too.

    We know p > 2

    3n + 1 is odd; therefore, 3n must be even

    If the above is prime, then n has to be even = 2k

    3n + 1 = 3(2k) + 1 = 3k + 1
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