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Re: [PrimeNumbers] Re: proving the Riemann hypothesis

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  • Joshua Zucker
    ... I think Dick thought you were proving that there s a prime between p^2 - p and p^2 + p. I think the rest of us thought you were proving that there s a
    Message 1 of 40 , Feb 2, 2007
      On 2/2/07, Shi Huang <shuangtheman@...> wrote:
      >
      > --- Dick <richard042@...> wrote:
      > > > I can prove that the gap must be smaller
      > > > than P1xP1-P1. If P1 is 5, this means that the gap
      > > > between 5 and its next prime must be smaller than
      > > 5 x
      > > > 5 -5=20.
      > >
      > > Much stronger than Bertrand's postulate and a very
      > > sensible conjecture
      > > heuristically.
      >
      > Dick,
      >
      > I am not a math specialist. So I was discouraged by
      > earilier replies to my post, which implies what I have
      > conjectured is similar to Bertrand's postulate. Could
      > you please explain why you think it is much stronger?

      I think Dick thought you were proving that there's a prime between p^2
      - p and p^2 + p.
      I think the rest of us thought you were proving that there's a prime
      between p and p^2 - p.

      > I now also realized that my conjecture is not enough
      > to prove the RH. It mearly says that the gap is not
      > completely random but it does not say that the gap is
      > as non-random as possble which is what the RH means.

      I think that these kinds of very rough paraphrases are not very useful.

      > 2. prove that primes are generated by a precise law,
      > where no randomness is involved.
      Of course there's no randomness involved -- a number is either prime
      or it isn't, and we can compute that.

      The question is in what ways the sequence of primes "looks like" a
      random sequence, by which we mean that properties like the length of
      the shortest gap in a given range of numbers, or the longest gap, or
      the percent prime, might be similar to a "probabilistically-generated
      prime-like" sequence.

      > I believe I can prove both 1 and 2. Would this be
      > enough to prove the RH?

      No.

      > If the
      > essence of primes says that prime is unpredictable,
      > then of course primes should give a chance appearance.

      What makes primes so interesting is that they are predictable and yet
      still in many ways give a chance appearance: the sequence of primes
      has many, many properties in common with a probabilistically-generated
      sequence.

      > If the essence also says that primes are 100%
      > lawfully generated where randomness is completely
      > ruled out, then of course, primes would show extreme
      > regularity.

      But what exactly do you mean by "regularity"?

      Primes are of course 100% lawfully generated and there is no randomness.

      --Joshua Zucker
    • Werner D. Sand
      For example 2 adjacent gaps cannot be equal if they aren t multiple of 6. For example the gap between 2 pairs of twins is at least 4. For example each prime
      Message 40 of 40 , Feb 7, 2007
        For example 2 adjacent gaps cannot be equal if they aren't multiple of
        6. For example the gap between 2 pairs of twins is at least 4. For
        example each prime number has the form 2n+/-1, 3n+/-1, 4n+/-1, 6n+/-1.
        Each pair of twins has the form 12n+-1, there are approximate formulas
        for the nth prime and the number of primes < x and so on. You cannot
        call all this random ore unpredictable. Of course the prime numbers are
        distributed as regularly as possible, that's a tautology. In
        mathematics everything is as regular as possible. Is pi random? Build
        P=2,357111317192329…, and you have the same case as pi. Consider the
        primes to be an irrational number, and there are no problems. If you
        mean there is no formula f(n) which produces primes for each n, then
        you are right. In this sense primes are random. (I am not quite sure –
        there is a formula p=[k^n^3] (H.W.Mills) which is said to produce only
        prime numbers). If you define "formula" as an algorithm, as a
        calculation instruction such as the sieve of Eratosthenes, then the
        primes are not random but simply what they are. Perhaps the compound
        numbers are random? Or are they only non-transparently complicated?

        Werner
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