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Extension of Dirchlet Theorem

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  • Adam
    RE: Adam Karasek post for pythagorean triples with a and c primes and b a product of at almost four primes, I seek primes p with 2p-1 and 4p-1 simlutaneously
    Message 1 of 2 , Nov 1, 2003
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      RE: Adam Karasek post for pythagorean triples with a and c primes and
      b a product of at almost four primes, I seek primes p with 2p-1 and
      4p-1 simlutaneously prime (to reformulate, we could write p=2n+1, and
      then seek 2n+1,4n+1,8n+1 simultaneously prime). I am having poor
      luck finding such p values, contrary to the belief in the extension
      of Dirichlet's theorem. For instance, I generate a random 12 digit
      prime, search the through the next 5000 primes, and find about 200-
      300 that have 2p-1 prime, but only 0 of those having 4p-1 prime.

      Am I not being patient enough or...?

      Adam
    • Lawrence Hon
      I dont think any exist besides a few trivial cases. First, p must be congruent to either 1 or 2 mod 3 because if it was congruent to 0 it would be a multiple
      Message 2 of 2 , Nov 1, 2003
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        I dont think any exist besides a few trivial cases.

        First, p must be congruent to either 1 or 2 mod 3 because if it was
        congruent to 0 it would be a multiple of 3.

        so we have two cases:

        case 1:
        p = 1 mod 3
        2p-1 = 1 mod 3
        4p-1 = 3 mod 3 = 0, so 4p-1 is a multiple of 3 (not prime)

        case 2:
        p = 2 mod 3
        2p-1 = 3 mod 3 = 0 so 2p -1 is a multiple of 3 (not prime)
        4p-1 = 5 mod 3 = 2

        thus, this never works unless we take 3 to be that multiple.

        so this implies in case 1, p = 1 (not a solution)
        or in case 2, 2p - 1 = 3, p = 2
        so p = 2, 2p-1 = 3, 4p-1 = 7 so (2,3,7) is a solution

        lastly, p could be = 0 mod 3, so p = 3, 2p-1 = 5, 4p-1 = 11
        so (3,5,11) is also a solution.

        Thus there are only two solutions and your search is futile :)

        Lawrence




        On Sat, 1 Nov 2003, Adam wrote:

        > RE: Adam Karasek post for pythagorean triples with a and c primes and
        > b a product of at almost four primes, I seek primes p with 2p-1 and
        > 4p-1 simlutaneously prime (to reformulate, we could write p=2n+1, and
        > then seek 2n+1,4n+1,8n+1 simultaneously prime).  I am having poor
        > luck finding such p values, contrary to the belief in the extension
        > of Dirichlet's theorem.  For instance, I generate a random 12 digit
        > prime, search the through the next 5000 primes, and find about 200-
        > 300 that have 2p-1 prime, but only 0 of those having 4p-1 prime.
        >
        > Am I not being patient enough or...?
        >
        > Adam
        >
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