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RE: [PrimeNumbers] Euclidean generation of prime numbers

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  • Bhupinder Singh Anand (R)
    On Thursday, February 01, 2007 4:17 AM, Alastair Farrugia wrote in primenumbers@yahoogroups.com: AF ... Is there any literature about the properties of this
    Message 1 of 40 , Jan 31, 2007
      On Thursday, February 01, 2007 4:17 AM, Alastair Farrugia wrote in
      primenumbers@yahoogroups.com:

      AF>> ... Is there any literature about the properties of this algorithm
      ... <<AF

      Alastair
      =======
      Years ago, I investigated similar sieve algorithms, which you can find
      at:

      Three Theorems on modular sieves that suggest the Prime Difference is
      O(Number of primes < (p(n)^1/2))

      http://alixcomsi.com/Three_Theorems.htm
      http://alixcomsi.com/Three_Theorems.pdf

      Regards,

      Bhup
    • 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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