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Mersenne Sum Converges

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  • Sebastian Martin
    Hello all: I send you a result about mersenne prims exponents: Mersenne Sum Converges Let Mn=nth mersenne prime number. Let mn=nth Mersenne exponent
    Message 1 of 1 , Aug 27, 2006
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      Hello all:

      I send you a result about mersenne prims exponents:

      Mersenne Sum Converges

      Let Mn=nth mersenne prime number.

      Let mn=nth Mersenne exponent 2^mn-1=Mn

      We have: log_2log_2Mn =~ 0.5649n+0.8255
      see
      http://primes.utm.edu/notes/faq/NextMersenne.html

      Then log_2mn =~0.5649n+0.8255

      If we consider Sum[1/mn^gamma,{n,1,infty}]

      1/mn^gamma =~ 1/(2^(0.5649n+0.8255) )^gamma
      <1/2^(0.3n)

      Sum[1/2^0.3n] converges, then
      Sum[1/mn^gamma,{n,1,infty}] converges to a real
      number alpha.

      If we sum only the known terms:
      Sum[1/mi^eulergamma,{i,1,43}]=3.10965012702975

      Question:

      Is alpha=Pi ????

      See bellow the convergence:

      A={2,3,5,7,13,17,19,31,61,89,107,127,521,
      607,1279,2203,2281,3217,4253,4423,
      9689,9941,11213,19937,21701,23209,44497,86243,110503,132049,216091,
      756839,859433,1257787,1398269,2976221,3021377,6972593,13466917,
      20996011,24036583,25964951,30402457}

      Length[A]
      43

      N[EulerGamma,15]
      0.577215664901533

      S[n_]:=N[Sum[1/A[[j]]^EulerGamma,{j,1,n}],15]
      Do[Print[n," ",S[n]],{n,1,43,1}]

      1 0.670256093380729
      2 1.20064952571047
      3 1.59560084074936
      4 1.92083501481544
      5 2.14835232635520
      6 2.34323158001754
      7 2.52599252290907
      8 2.66376506388321
      9 2.75697874352069
      10 2.83193045448214
      11 2.89932238438207
      12 2.96036758655390
      13 2.98739464041434
      14 3.01214034871067
      15 3.02823441007535
      16 3.03999311965759
      17 3.05151802801339
      18 3.06096829990708
      19 3.06901207986911
      20 3.07687592718745
      21 3.08187693280040
      22 3.08680436596281
      23 3.09140097033442
      24 3.09469835037347
      25 3.09783825140524
      26 3.10085872276354
      27 3.10293320961256
      28 3.10434907796057
      29 3.10557619357864
      30 3.10668340733983
      31 3.10751663588951
      32 3.10792079007793
      33 3.10829635053597
      34 3.10859779771992
      35 3.10888137325548
      36 3.10906473091244
      37 3.10924650175030
      38 3.10935867404387
      39 3.10943538806101
      40 3.10949475544925
      41 3.10954966458904
      42 3.10960218152565
      43 3.10965012702975

      N[Pi,15]
      3.14159265358979




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