## Finding Large PRP's in Parallel

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• Prob( y
Message 1 of 1 , Jul 29 11:19 PM
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Prob( y < g ) = 1 - exp( -g / (ln z - ln ln z) )
for gap of size g for primes around number z
{Kulsha 7-26-01}

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,
so testing
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
mitlbrown@...

----- 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

> Hello!
>
> Andrey Kulsha wrote:
>
> > General result: the probability of a given gap g around the number x
being
> > 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,
>
> Andrey
>
>
>
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