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Re: [PrimeNumbers] Re: Fw: A new conjecture on primes

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  • Peter Kosinar
    ... Ignoring Sun s definition of P_i, if we set = , the corresponding value of w_1(3) would be equal to 4; not a prime. Thus, there must
    Message 1 of 10 , Mar 21, 2012
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      > > CONJECTURE ON PRIMES (Z. W. Sun, March 17-18, 2012). For k=1,2,3,... let
      > > P_k denote the product of the first k primes p_1,...,p_k.
      > >
      > >   (i) For n=1,2,3,... define w_1(n) as the least integer m>1 such that m
      > > divides none of P_i-P_j with i,j distinct and not more than n. Then
      > > w_1(n) is always a prime.
      >
      > Isn't it self evident that the first (least) integer which doesn't divide into another integer, is a prime number? Or am I misunderstanding something here ...

      Ignoring Sun's definition of P_i, if we set <P1, P2, P3> = <1, 3, 4>, the
      corresponding value of w_1(3) would be equal to 4; not a prime. Thus,
      there must be something special about his definition of P_i, which makes
      the resulting ones primes [or his conjecture is false :-) ].

      Peter

      [Non-text portions of this message have been removed]
    • jbrennen
      ... Hint: what is the least integer which does not divide 6? It s evident that the least integer in question is a prime power, but it s not necessarily a
      Message 2 of 10 , Mar 21, 2012
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        --- In primenumbers@yahoogroups.com, "Mark" <mark.underwood@...> wrote:
        >
        >
        > Isn't it self evident that the first (least) integer which doesn't divide into another integer, is a prime number? Or am I misunderstanding something here ...
        >
        >
        > Mark
        >

        Hint: what is the least integer which does not divide 6?

        It's evident that the least integer in question is a prime power, but it's not necessarily a prime.
      • Mark
        True, true, a prime power, not simply a prime. Now I suppose the question is, why hasn t there been a counterexample to Sun s conjectures. I m guessing it is
        Message 3 of 10 , Mar 21, 2012
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          True, true, a prime power, not simply a prime. Now I suppose the question is, why hasn't there been a counterexample to Sun's conjectures. I'm guessing it is just because prime powers are relatively scarce compared to the primes.

          Mark



          --- In primenumbers@yahoogroups.com, "jbrennen" <jfb@...> wrote:
          >
          > --- In primenumbers@yahoogroups.com, "Mark" <mark.underwood@> wrote:
          > >
          > >
          > > Isn't it self evident that the first (least) integer which doesn't divide into another integer, is a prime number? Or am I misunderstanding something here ...
          > >
          > >
          > > Mark
          > >
          >
          > Hint: what is the least integer which does not divide 6?
          >
          > It's evident that the least integer in question is a prime power, but it's not necessarily a prime.
          >
        • Mark
          If I m not mistaken Sun is saying that the least number to not divide, say, 11# - 7#, is a prime. But the first number to not divide this is 8. Mark
          Message 4 of 10 , Mar 21, 2012
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            If I'm not mistaken Sun is saying that the least number to not divide, say, 11# - 7#, is a prime. But the first number to not divide this is 8.

            Mark




            --- In primenumbers@yahoogroups.com, "Mark" <mark.underwood@...> wrote:
            >
            >
            >
            >
            > True, true, a prime power, not simply a prime. Now I suppose the question is, why hasn't there been a counterexample to Sun's conjectures. I'm guessing it is just because prime powers are relatively scarce compared to the primes.
            >
            > Mark
            >
            >
            >
            > --- In primenumbers@yahoogroups.com, "jbrennen" <jfb@> wrote:
            > >
            > > --- In primenumbers@yahoogroups.com, "Mark" <mark.underwood@> wrote:
            > > >
            > > >
            > > > Isn't it self evident that the first (least) integer which doesn't divide into another integer, is a prime number? Or am I misunderstanding something here ...
            > > >
            > > >
            > > > Mark
            > > >
            > >
            > > Hint: what is the least integer which does not divide 6?
            > >
            > > It's evident that the least integer in question is a prime power, but it's not necessarily a prime.
            > >
            >
          • Peter Kosinar
            ... Nope -- Sun claims that for any N, the first number which does not divide ANY of the primorials A# - B# for A, B
            Message 5 of 10 , Mar 21, 2012
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              > If I'm not mistaken Sun is saying that the least number to not divide, say, 11# - 7#, is a prime. But the first number to not divide this is 8.
              >
              > Mark

              Nope -- Sun claims that for any N, the first number which does not divide
              ANY of the primorials A# - B# for A, B <= N, is a prime.

              Peter
            • Mark
              ... Ah, I see, thanks for the clarification. So as a counterexample we re looking for a prime power (with exponent greater than one) which is the least number
              Message 6 of 10 , Mar 21, 2012
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                --- In primenumbers@yahoogroups.com, Peter Kosinar <goober@...> wrote:
                >
                > > If I'm not mistaken Sun is saying that the least number to not divide, say, 11# - 7#, is a prime. But the first number to not divide this is 8.
                > >
                > > Mark
                >
                > Nope -- Sun claims that for any N, the first number which does not divide
                > ANY of the primorials A# - B# for A, B <= N, is a prime.
                >
                > Peter
                >

                Ah, I see, thanks for the clarification. So as a counterexample we're looking for a prime power (with exponent greater than one) which is the least number that does not divide any combination of A# - B# for A,B up to a given n. I think I can see why that should be next to impossible.



                > CONJECTURE ON PRIMES (Z. W. Sun, March 17-18, 2012). For k=1,2,3,... let
                > P_k denote the product of the first k primes p_1,...,p_k.
                >
                > (i) For n=1,2,3,... define w_1(n) as the least integer m>1 such that m
                > divides none of P_i-P_j with i,j distinct and not more than n. Then
                > w_1(n) is always a prime.
              • Mark
                ... No, I m wrong again. The least number to not divide another number will indeed be a prime power, but I m wrong to assume this would be true when applied
                Message 7 of 10 , Mar 21, 2012
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                  --- In primenumbers@yahoogroups.com, "Mark" <mark.underwood@...> wrote:
                  >
                  >
                  >
                  > --- In primenumbers@yahoogroups.com, Peter Kosinar <goober@> wrote:
                  > >
                  > > > If I'm not mistaken Sun is saying that the least number to not divide, say, 11# - 7#, is a prime. But the first number to not divide this is 8.
                  > > >
                  > > > Mark
                  > >
                  > > Nope -- Sun claims that for any N, the first number which does not divide
                  > > ANY of the primorials A# - B# for A, B <= N, is a prime.
                  > >
                  > > Peter
                  > >
                  >
                  > Ah, I see, thanks for the clarification. So as a counterexample we're looking for a prime power (with exponent greater than one) which is the least number that does not divide any combination of A# - B# for A,B up to a given n. I think I can see why that should be next to impossible.
                  >

                  No, I'm wrong again. The least number to not divide another number will indeed be a prime power, but I'm wrong to assume this would be true when applied to more than one number. The problem is more subtle and difficult than I supposed.


                  Mark



                  >
                  >
                  > > CONJECTURE ON PRIMES (Z. W. Sun, March 17-18, 2012). For k=1,2,3,... let
                  > > P_k denote the product of the first k primes p_1,...,p_k.
                  > >
                  > > (i) For n=1,2,3,... define w_1(n) as the least integer m>1 such that m
                  > > divides none of P_i-P_j with i,j distinct and not more than n. Then
                  > > w_1(n) is always a prime.
                  >
                • Jack Brennen
                  ... As would be expected when coming from Zhi-Wei Sun: http://en.wikipedia.org/wiki/Sun_Zhiwei If he presents it as a conjecture, you can be sure of two
                  Message 8 of 10 , Mar 21, 2012
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                    On 3/21/2012 3:56 PM, Mark wrote:
                    >
                    > No, I'm wrong again. The least number to not divide another number
                    > will indeed be a prime power, but I'm wrong to assume this would be
                    > true when applied to more than one number. The problem is more
                    > subtle and difficult than I supposed.
                    >

                    As would be expected when coming from Zhi-Wei Sun:

                    http://en.wikipedia.org/wiki/Sun_Zhiwei

                    If he presents it as a conjecture, you can be sure of two things...
                    It's very likely true, and will be very hard to prove.
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