## Mersenne Sum Converges

Expand Messages
• 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
• 0 Attachment
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

______________________________________________
LLama Gratis a cualquier PC del Mundo.
Llamadas a fijos y móviles desde 1 céntimo por minuto.
http://es.voice.yahoo.com
Your message has been successfully submitted and would be delivered to recipients shortly.