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Re: Fw: A new conjecture on primes

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  • Mark
    ... 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
    Message 1 of 10 , Mar 21, 2012
      > 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 ...


      Mark




      --- In primenumbers@yahoogroups.com, Steven Harvey <harvey563@...> wrote:
      >
      > I thought the group might find this interesting.
      >  
      > StevenHarvey
      >
      > harvey563@...
      >
      > ----- Forwarded Message -----
      > From: Zhi-Wei Sun <zwsun@...>
      > To: NMBRTHRY@...
      > Sent: Monday, March 19, 2012 6:24 PM
      > Subject: A new conjecture on primes
      >
      > Dear number theorists,
      >
      >   Prime numbers are very mysterious and there are many unsolved
      > problems on primes.
      >
      >   Here I report that I have formulated a new conjecture on primes which
      > allows us to produce primes via primes.
      >
      > 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.
      >
      >   (ii) For n=1,2,3,... define w_2(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_2(n) is always a prime.
      >
      >   (iii) We have w_1(n)<n^2 and w_2(n)<n^2 for all n=2,3,4,....
      >
      >   Clearly w_i(n) does not exceed w_i(n+1) for i=1,2. Since P_1,...,P_n
      > are pairwise distinct modulo w_1(n), we see that w_1(n) is at least n
      > and hence W_1={w_1(n): n=1,2,3,...} is an infinite set. For any integer
      > m>1, there is an odd prime p_n=-1 (mod m) by Dirichlet's theorem and
      > hence P_{n-1}+P_n=P_{n-1}(1+p_n) is a multiple of m. Thus W_2={w_2(n):
      > n=1,2,3,...} is also infinite.
      >
      >   If w_i(n) is the k-th prime p_k, then k is at least n since p_k
      > divides both P_{k+1}-P_k and P_{k+1}+P_k. Thus part (ii) of the
      > conjecture implies the inequality w_2(n)>n, which I am even unable to prove.
      >
      >   A prime is said to be of the first kind (or the second kind) if it
      > belongs to W_1={w_1(n): n=1,2,3,...} (or W_2={w_2(n): n=1,2,3,...},
      > resp.). Here I list the first 20 primes of each kind.
      >
      > Primes of the first kind:  2, 3, 5, 11, 23, 29, 37, 41, 47, 73, 131,
      > 151, 199, 223, 271, 281, 353, 457, 641, 643, ...
      >
      > Primes of the second kind: 2, 3, 5, 7, 11, 19, 23, 47, 59, 61, 71, 101,
      > 113, 223, 487, 661, 719, 811, 947, 1327, ...
      >
      >   I have added the two sequences w_1(1),w_1(2),w_1(3),...  and
      > w_2(1),w_2(2),w_2(3),... to OEIS, see
      > http://oeis.org/A210144%c2%a0 and  http://oeis.org/A210186
      > where you can find values of w_1(n) for n=1,...,1172 and values of
      > w_2(n) for n=1,...,258.
      >
      >   In my opinion, the above conjecture might be very challenging and it
      > contains certain mysterious information about primes. It seems that
      > parts (i) and (ii) of the conjecture remain true if we replace P_k by
      > (P_k)^d or (-1)^k*(P_k)^d where d is any positive integer.
      >
      >   Any comments are welcome!
      >
      >
      >   Zhi-Wei Sun
      >   http://math.nju.edu.cn/~zwsun
      >
      > [Non-text portions of this message have been removed]
      >
    • 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 2 of 10 , Mar 21, 2012
        > > 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 3 of 10 , Mar 21, 2012
          --- 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 4 of 10 , Mar 21, 2012
            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 5 of 10 , Mar 21, 2012
              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 6 of 10 , Mar 21, 2012
                > 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 7 of 10 , Mar 21, 2012
                  --- 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 8 of 10 , Mar 21, 2012
                    --- 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 9 of 10 , Mar 21, 2012
                      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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