Thread for posting tiny primes - Page 54

science_man_88 · 2010-09-22, 13:06

okay so I have to program in

(((t/log(N))^n)*exp(t/log(N)))/n! that shouldn't be too hard.

science_man_88 · 2010-09-22, 13:09

Code:

poisson2(N,t,n) = (((t/log(N))^n)*exp(t/log(N)))/n!

CRGreathouse · 2010-09-22, 13:16

I think something's wrong: it gave a 109% chance for 20 numbers near a googol to have 0 primes.

science_man_88 · 2010-09-22, 13:19

I found it lol I forgot the - sign

science_man_88 · 2010-09-22, 13:21

Code:

poisson2(N,t,n) = (((t/log(N))^n)*exp(-(t/log(N))))/n!

this is how it should of been lol I'll test it out again lol.

science_man_88 · 2010-09-22, 13:23

Code:

(10:21) gp > poisson2(10^100,0,20)
%89 = 0

this more likely ? I've got to increase my precision though.

CRGreathouse · 2010-09-22, 14:04

Quote:

Originally Posted by science_man_88

Code:

(10:21) gp > poisson2(10^100,0,20)
%89 = 0

this more likely ? I've got to increase my precision though.

That's the correct answer, but not the same as what I asked. It says that if you choose 0 numbers near a googol, your chances of finding 20 primes is 0.

For my question, I get

Code:

poisson2(1e100,20,0)
%1 = 0.9168064512151768221426797071

science_man_88 · 2010-09-22, 14:24

Quote:

Originally Posted by CRGreathouse

That's the correct answer, but not the same as what I asked. It says that if you choose 0 numbers near a googol, your chances of finding 20 primes is 0.

For my question, I get

Code:

poisson2(1e100,20,0)
%1 = 0.9168064512151768221426797071

sorry I'm getting confused lol.

science_man_88 · 2010-09-22, 14:42

if I did the math correct that means 230 numbers near a googol seems to give the best chance of exactly 1 being prime at 36.7879 % I think

CRGreathouse · 2010-09-22, 14:47

No problem.

So first, a warning: choosing 20 numbers isn't really a good example, since 20 is "small" and so the binomial distribution would be more appropriate than the Poisson distribution. (Conceivably, a program could switch between these depending on input parameters; for now let's keep things simple and leave it using only Poisson.)

Moving on: a number that is known to have no prime factors below L is approximately $e^\gamma\log L$ times as likely to be prime as a 'normal' number, assuming L isn't too close to the square root of the number. Use this to modify the program by adding a parameter L that changes the lambda appropriately. Note that $\gamma$ is

Code:

Euler

in Pari.

Mini-Geek · 2010-09-22, 14:49

Quote:

Originally Posted by science_man_88

if I did the math correct that means 230 numbers near a googol seems to give the best chance of exactly 1 being prime at 36.7879 % I think

Which makes sense, since log(10^100) is about 230, and about 1 in x numbers the size of N are prime where log(N)=x.

2010-09-22, 13:09	#585
science_man_88 "Forget I exist" Jul 2009 Dumbassville 10000011000000₂ Posts	Code: poisson2(N,t,n) = (((t/log(N))^n)exp(t/log(N)))/n! Last fiddled with by science_man_88 on 2010-09-22 at 13:09*

2010-09-22, 13:21	#588
science_man_88 "Forget I exist" Jul 2009 Dumbassville 2⁶·131 Posts	Code: poisson2(N,t,n) = (((t/log(N))^n)*exp(-(t/log(N))))/n! this is how it should of been lol I'll test it out again lol.

2010-09-22, 13:23	#589
science_man_88 "Forget I exist" Jul 2009 Dumbassville 8384₁₀ Posts	Code: (10:21) gp > poisson2(10^100,0,20) %89 = 0 this more likely ? I've got to increase my precision though.

2010-09-22, 14:47	#593
CRGreathouse Aug 2006 3·1,993 Posts	No problem. So first, a warning: choosing 20 numbers isn't really a good example, since 20 is "small" and so the binomial distribution would be more appropriate than the Poisson distribution. (Conceivably, a program could switch between these depending on input parameters; for now let's keep things simple and leave it using only Poisson.) Moving on: a number that is known to have no prime factors below L is approximately $e^\gamma\log L$ times as likely to be prime as a 'normal' number, assuming L isn't too close to the square root of the number. Use this to modify the program by adding a parameter L that changes the lambda appropriately. Note that $\gamma$ is Code: Euler in Pari. Last fiddled with by CRGreathouse on 2010-09-22 at 14:48

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2010-09-22, 13:06	#584
science_man_88 "Forget I exist" Jul 2009 Dumbassville 2⁶·131 Posts	okay so I have to program in (((t/log(N))^n)*exp(t/log(N)))/n! that shouldn't be too hard.

2010-09-22, 13:16	#586
CRGreathouse Aug 2006 3×1,993 Posts	I think something's wrong: it gave a 109% chance for 20 numbers near a googol to have 0 primes.

2010-09-22, 13:19	#587
science_man_88 "Forget I exist" Jul 2009 Dumbassville 20300₈ Posts	I found it lol I forgot the - sign

2010-09-22, 14:42	#592
science_man_88 "Forget I exist" Jul 2009 Dumbassville 2⁶·131 Posts	if I did the math correct that means 230 numbers near a googol seems to give the best chance of exactly 1 being prime at 36.7879 % I think