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Prime Formulas

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  • Sebastian Martin Ruiz
    Hello all: I ve gotten all these formulas are verified for ALL Prime Numbers based on the harmonic numbers. Pn = Floor [H [(1 +1 / Pn) ^ ((Pn) ^ 2)]-EulerGamma
    Message 1 of 1 , Oct 29, 2011
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      Hello all:

      I've gotten all these formulas are verified for ALL Prime Numbers based on the harmonic numbers.

      Pn = Floor [H [(1 +1 / Pn) ^ ((Pn)
      ^ 2)]-EulerGamma +3/2 - n * (log n) / (Pn)
      -1/n]

      For all n> = 1, ie equality is true for all
      primes.

      Other formulas that
      I have obtained are as follows:

      Pn =Floor [H [(1 +1 / Pn) ^ ((Pn)
      ^ 2) / E ^ ((log n) +EulerGamma-(Pn)
      /n -1/2 +3/(2n))]

      And this one:

      Pn = Floor [H [(1 +1 / Pn) ^ ((Pn)
      ^ 2) / E ^ (H
      [n] - (Pn) / n-1 / 2 +1 / n)]

      The three are verified for all prime numbers.

      All of them are based on the following result which I obtained:

      H [(1 +1 / n) ^ (n ^ 2)] = n + EulerGamma - 1/2 +Epsilon  with Epsilon -> 0 for all sufficiently large positive integer n. But ndoes not
      need tobevery large so that the convergence is
      good.


      Floor [x] = The higest integer equal
      or less than x.

      EulerGamma = 0.57721566 .... Euler Mascheroni constant.

      E = 2.71828182845 ... ...

      Pn = nth prime number.

      Log [n] = Ln [n]
       
      H[x]= HarmonicNumber[x]
       
       
      MATHEMATICA FILE:
       
       
      F[n_]:=Floor[HarmonicNumber[(1+1/(Prime[n]))^(Prime[n]^2)/E^(HarmonicNumber[n]-Prime[n]/n-1/2+1/n)]]
       G[n_]:=Floor[HarmonicNumber[(1+1/(Prime[n]))^(Prime[n]^2)/E^(Log[n]+EulerGamma-Prime[n]/n-1/2+3/(2n))]]
       
      FG[n_]:=Floor[HarmonicNumber[(1+1/Prime[n])^((Prime[n])^2)]-EulerGamma+3/2-n
      Log[n]/Prime[n]-1/n]
       Do[Print[FG[n]," ",F[n],"
      ",G[n]," ",Prime[n]],{n,1,50,1}]
       2   2   2   2
       3   3   3   3
       5   5   5   5
       7   7   7   7
       11   11   11   11
       13   13   13   13
       17   17   17   17
       19   19   19   19
       23   23   23   23
       29   29   29   29
       31   31   31   31
       37   37   37   37
       41   41   41   41
       43   43   43   43
       47   47   47   47
       53   53   53   53
       59   59   59   59
       61   61   61   61
       67   67   67   67
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       73   73   73   73
       79   79   79   79
       83   83   83   83
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       127   127   127   127
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       223   223   223   223
       227   227   227   227
       229   229   229   229

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