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proving the Riemann hypothesis

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  • Shi Huang
    The Riemann Hypothesis essentially says that the primes are as regularly distributed as possible. The mathematician Enrico Bombieri put it: “If the Riemann
    Message 1 of 40 , Feb 1, 2007
      The Riemann Hypothesis essentially says that the
      primes are as regularly distributed as possible. The
      mathematician Enrico Bombieri put it: “If the Riemann
      Hypothesis turns out to be false, there will be huge
      oscillations in the distribution of primes. In an
      orchestra, that would be like one loud instrument that
      drowns out the others - an aesthetically distasteful
      situation."

      If the RH is false, the gap between prime P1 and the
      next prime P2 could be anywhere from 1 to infinitely
      large. The gap would be a random number from 1 to
      infinity. To prove the RH true, we only need to prove
      that the gap is not without a consistent or fixed
      upper limit. I have found a simple proof for this
      that could be understood by anyone above the level of
      elementary school. I am writing it up for
      publication. 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.

      Can anyone else also prove this? If this can be
      proven, would it represent a proof of the RH? The RH
      is merely a special manifestation of the regularity of
      primes. This regularity could be manifestated in
      other ways. A proof for any one of these
      manifestations would prove the regularity of primes,
      which in turn would take care of all other
      manifestations. The regularity of primes is the
      foundation, which give rise to the RH. Any proof for
      the regularity is a proof for the RH. A direct proof
      of the specific statement of RH (all zeros are on one
      line) may indeed be impossible. It may have to be
      proved from a totally different perspective. As
      Einstein put it, a problem cannot be solved by the
      same mindset that created the problem.

      I would further conjecture that the gap must be
      smaller than the value of P1, but I have not been able
      to prove it.





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    • 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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