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#1 2006-02-18 22:44:56

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Sum of the reciprocals of the squares of the prime numbers

Very interesting prblem:


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#2 2006-02-18 22:46:53

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: Sum of the reciprocals of the squares of the prime numbers

We know that:


So
0<Sum<Pi^2/6


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#3 2006-02-18 22:48:35

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: Sum of the reciprocals of the squares of the prime numbers

I'll try to make a program for the sum.


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#4 2006-02-18 22:51:33

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Sum of the reciprocals of the squares of the prime numbers

Here's my program:

k = 100000;
ss = 1000;
Print["Initialization Done"];
Do[prt[i] = Prime[i], {i, 1, k}];
Print["Prime Geneneration Done"];
Do[prtsq[i] = prt[i]*prt[i], {i, 1, k}];
Print["Squaring Done"];
Do[thearr[i] = 1/prtsq[i], {i, 1, 100000}];
Print["Factorising Done"];
sum = 0;
Do[sum += N[thearr[i], ss], {i, 1, 100000}];
Print["Sum = ", sum];

And here's what I get for the sum:
0.4522473688277700954422164277522326096645066956889359406373748543156987039966
525643794108225095161389758244363445339660776113109242885808492718205416638194
526958800379786519866057949036856229177825278987902389864970540352844167196863
834688645075159015805935496856212223538491039364902103635445065361656745970932
905021125274435235568533828552570131338540212094414196757580439057169807537575
899440924188480149221835203239871132457344235369007243117411236106046262308444
576868350724155932530725321698618605264299442562165096782086212268498382688102


IPBLE:  Increasing Performance By Lowering Expectations.

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#5 2006-02-18 22:52:49

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Sum of the reciprocals of the squares of the prime numbers

Is this sum a irrational number?
Is It transcedent?
Can we compute Prime[n] from it?


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#6 2006-02-20 09:09:29

John E. Franklin
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Registered: 2005-08-29
Posts: 3,562

Re: Sum of the reciprocals of the squares of the prime numbers

I get everything except the N and the ss usage.


igloo myrtilles fourmis

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#7 2006-02-22 02:54:48

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: Sum of the reciprocals of the squares of the prime numbers

In Mathematica the "N(number,precise)" function gives the number with precisedigits after the demical point.
Here's an example:
N[Pi,2]=3.14
N[Pi,4]=3.1415
ss is the precise. Because ss=1000 we can be sure that more than 100 digits or this number are correct.


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#8 2006-02-22 05:55:10

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,562

Re: Sum of the reciprocals of the squares of the prime numbers

And if Prime[] returns 0 if not prime, then why don't you get divide by zero error?
Tell me about Prime[]
Does prime return infinitywhen not prime?


igloo myrtilles fourmis

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#9 2006-02-22 09:15:05

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,534

Re: Sum of the reciprocals of the squares of the prime numbers

Maybe we could it "Krassi's Number"


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#10 2006-02-28 06:15:37

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Sum of the reciprocals of the squares of the prime numbers

For John, not, PrimeQ[] is for testing whatever a number is prime.
Prime[n] gives the n-th pime number:
Prime[1]=2
Prime[2]=3
Prime[3]=5
...


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#11 2006-02-28 06:17:14

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Sum of the reciprocals of the squares of the prime numbers

And for rob-ya, it's actually a great idea to call it "Krassi's number". But it's not os scientific as "Geogiev's constant".


IPBLE:  Increasing Performance By Lowering Expectations.

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#12 2006-03-01 07:33:03

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,562

Re: Sum of the reciprocals of the squares of the prime numbers

Okay, now the program makes sense, thanks a lot.


igloo myrtilles fourmis

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#13 2006-03-02 11:17:52

God
Member
Registered: 2005-08-25
Posts: 59

Re: Sum of the reciprocals of the squares of the prime numbers

Maybe the solution lies in the infinite pi-notation format of the reimann zeta function
After all, problems are a lot cooler to solve by hand than to approximate with computer tongue

Last edited by God (2006-03-02 11:18:20)

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#14 2006-03-03 00:15:49

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Sum of the reciprocals of the squares of the prime numbers

What is that infinite pi-notation?
Can you explain it...
Simply?


IPBLE:  Increasing Performance By Lowering Expectations.

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