Finding Large PRP's in Parallel
- Prob( y < g ) = 1 - exp( -g / (ln z - ln ln z) )
for gap of size g for primes around number z
If Prob = 50% and z = 10^x + c, small c, then
.5 = exp ( -g / ( x ln 10 - ln( x ln 10) )
g = ln 2 ( x ln 10 - ln(x ln 10) )
g = 0.69315 ( 2.302585 x - ln (2.302585 x) )
g = 1.596 x - 0.69315 ln(2.302585 x)
g ~ 1.6 x
On the average 10^x will be half way into the gap,
10^x + 1, 3, ... , 0.8x
10^(x+1) + 1, 3, ... , 0.8(x+1)
10^(x+2) + 1, 3, ... , 0.8(x+2)
10^(x+3) + 1, 3, ... , 0.8(x+3)
gives 50% probabilty of finding a prime for each.
If these are run in parallel, on 4 different computers,
you will have a 93.75% probability of finding a PRP
at the end of the search. (With 5 computers about 97%)
Milton L. Brown
----- Original Message -----
From: "Andrey Kulsha" <Andrey_601@...>
Cc: "Milton Brown" <miltbrown@...>
Sent: Thursday, July 26, 2001 9:30 AM
Subject: Re: [PrimeNumbers] Gaps distribution: Conjecture
> Andrey Kulsha wrote:
> > General result: the probability of a given gap g around the number x
> > greater than G is about
> > exp( -g / (log x + log(log x)) ).
> Sorry, exp( -g / (log x - log(log x)) ) is right.
> Therefore 436 is right in Milton's example instead of 433.
> However, log log x is too small, so
> exp( -g / log x)
> is good enough.
> Best wishes,
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