Re: Primes and Probability

From: Spike Jones (
Date: Sat May 12 2001 - 17:49:30 MDT

> Spike wrote: suppose you pick two random primes and
> > multiply them together to get a composite C. What is the probability
> > that C+2 is prime? Is it still 1/ln(C+2) ?
> > The answer to this is not trivial, and it bears directly on the
> > discussion on page 87 of Damien's updated version of The Spike. spike

Lee Corbin wrote:

> I'm lucky to have that book, wherein on page 87, it is related
> that a certain ominous **Mr. Jones** has determined the rate at
> which very large prime numbers are likely to be discovered.
> Now I would think that surely the probability that C+2 is
> prime is indeed proportional to 1/ln(C+2). Yes? It would be
> marvelous if someone had found otherwise. What's the connection? Lee

Ominous? Cooool, Ive always wanted to be ominous. Every time
you hear of an ominous donor giving money to something, you'll
know that was me.

Im leading you to an apparent paradox, Lee. The reason I mention
the C+2 problem it is this will lead you to the following: Since
we have no reason to believe that the probability of C+2 is anything
other than 1 / ln(C+2), lets assume it so. Now open a spreadsheet
to make this easier. Get your two randomly chosen primes (neither
of which can be 2), multiply them together to generate the composite C.
Now fill down, C+2, C+4, and so on. In the next column calculate
1 / ln {the first column}. The second column is the probability of
each number to its left being prime. Now calculate the cumulative
probability of C+2 and C+4 prime, then C+6 also prime, and so on.

What did you find out?

One of the many fascinating aspects of living when we do is that for
the first time in all human history, it is possible to make a prediction
about when the record for the largest known prime will be broken. To
do this, all you need to know is the following:

1. Now and for all future time, the largest known prime will always
be a Mersenne prime, since there is no known way nor any anticipated
discovery that will allow megadigit non-Mersenne numbers to be tested
for primality.

2. You need to know the status of the GIMPS project, the Great
Internet Mersenne Primes Search.

3. You need to understand how to calculate the probability of a given
number of being prime, given that it is of the form 2^P-1 where P is

4. You also understand how to adjust the probability if the number has
already been checked for small factors {"small" being defined as factors
smaller than about 6 octillion}.

When making the estimate for Damien's book, we had to make several
assumptions regarding the future of GIMPS, all of which were done
by extrapolation in May 2000. We extrapolated the growth in
the number of participants (which we assumed would grow linearly)
the speedup of processors (Moore's law with a 2 yr doubling time),
the continued steady improvement of the algorithm, plus several other
assumptions which have proved fairly accurate with two notable exceptions:

a) I failed to foresee the People's Republic of Taxifornia's power crisis

b) I did not account for the fact that George Woltman, the founder of
GIMPS {may he live forever and fill the earth with his descendants}
would continue to *optimize* the Prime95 fourier transforms for
*specific processors*. George is pending his time doing this for nothing
instead of taking that skill and making about a jillion dollars with it,
which proves Eliezer's point that there are a few people in this big
world who do things like this.

an ominous donor, spike

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