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Re: [PrimeNumbers] Re: proving the Riemann hypothesis

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  • Shi Huang
    ... Thanks for the clarification. ... Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language
    Message 1 of 40 , Feb 2, 2007
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      > 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.

      Thanks for the clarification.
      >
      > > 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.

      "Most of the fundamental ideas of science are
      essentially simple, and may, as a rule, be expressed
      in a language comprehensible to everyone."
      Albert Einstein (1879-1955)

      If we cannot prove the RH's exact statement, we may
      try to prove its simple and layperson statement. The
      RH is widely regarded as the most fundamental problem
      of math, if not mankind. Its basic idea should be
      comprehensible to everyone. If it is wrong, it should
      strike everyone as unreasonable. The thing about
      primes that bugs people and makes it uniquely
      interesting is the two opposite yin and yang facts as
      commented by Don Ziger: being both random and
      non-random. Neither facts are however proven and are
      only seemingly so. The RH supports the non-random
      aspect of primes. If primes lacks randomness, its
      non-randomness and in turn RH would be much less
      interesting. It is the seemingly impossible unity of
      two complete opposites that makes the prime so
      interesting.

      If the generation of primes, by the sieve method or
      the division method, can be chacterized or proven as
      non-random, we have at best proven only half of the
      picture. We must also prove that prime is random.
      What random means depends on how you want to prove it.
      The simplest is to prove that primes are
      unpredictable and no one has been able to do this.

      >
      > > 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.

      I am talking about a law that assumes we know nothing
      about the divisibility of primes. We know only that
      prime has one property which makes it both
      unpredictable as well as lawful or regular. The
      divisibility property of primes may make prime lawful
      but does not make or has not been proven to make prime
      unpredictable. It is easy to prove either the yin or
      yang alone, but it is hard to prove both yin and yang
      with the same stone. We need to find one property or
      essence of primes that takes care of both yin and
      yang.
      >
      > 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.

      I think I am getting at the core of the primes that is
      even more fundamental than the RH. The core is the
      yin and yang unity of random and non-random. we need
      to find a prove or reason for that unity. The RH is
      merely one of the two poles of the bipolar unity.
      Proving the RH alone would still not be enough to
      explain the unity. We still need to prove the
      randomness, which we can accomplish by proving
      unpredicatibility. So, I suggest a conjecture that
      says that there is a law that can generate individual
      prime precisely (non-random/lawfulness) but also
      demands that prime is unpredictable
      (random/lawlessness). Proving this conjecture or
      finding this law, in my view, takes care of the
      yin-yang duality that makse prime so interesting, and
      is in turn much more important than proving the RH.

      >
      > > 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

      I think you are wrong here. Primes are unpredictable.
      The total numbers under a number may be but not
      individual prime. If you cannot predict individual
      prime, then prime is unpredictable but this needs to
      be proven either false or true.

      What makes prime so interesting is the unity of random
      and non-random. But you confused non-random with
      predictability of total number of primes under a
      number. If random means unpredictability by way of a
      law or lawlessness, then its opposite must mean
      precise generation by a law or lawfulness. the same
      law that generates primes lawfully must in the mean
      time says that primes are unpredictable or random. No
      such law exists today. The sieve method or the
      division method may qualify as laws in precisely
      generating primes but they do not say or prove that
      primes must be random or unpredictable.

      Indeed, any law or conjecture that proves that primes
      are unpredictable would be a major breakthrough. It
      would save many people from wasting time trying to
      find a formula to predict individual prime.

      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"?

      following laws and being non-random.

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

      as I said earlier, the law that generates primes
      precisely must at the same time entails randomness to
      primes. No such law exists.

      We need to find a prime generating law that alone
      produces the random/lawlessness and
      non-random/lawfulness property of primes, which is the
      core of what makes prime so interesting, much more so
      than the RH which is merely one manifestation of the
      core. The RH, if proven true, proves the
      non-randomness of primes. but for that to be
      interesting, we must prove that primes are also
      random. The fact that people are still looking for
      formula to predict individual primes is plain evidence
      that no one has proven that primes are indeed random
      or unpredictable. They just seem to be random or show
      many properties of randomness. But we still need a
      proof to be sure.

      The universe seems to be randomly created, and most
      scientists believe it is. But there is no proof. The
      universe also seem to follow laws. So the universe
      (and all physical reality) is the closest thing to the
      primes, display a unity of random and non-random. It
      is easy to have laws that only describe the random
      aspect. It is also easy to have laws that only
      describe the non-random aspect. But what we need is a
      single law that explains or proves both the random and
      non-random aspects. Indeed, solving the prime puzzle
      may automatically take care of the deepest puzzle of
      the universe.

      Shi




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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
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        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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