# [ExI] 2^57885161-1

spike spike66 at att.net
Wed Feb 6 05:15:43 UTC 2013

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On Behalf Of Anders Sandberg
Subject: Re: [ExI] 2^57885161-1

So, Spike, how does this fit with current GIMPS trends?

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Anders Sandberg,

Thank you for asking, Anders.  What a delightful question!

I looked at doubling times of the largest known prime number, focusing on
only since 1950, because the largest known prime has almost always been a
Mersenne prime since those days, and will be for the remainder of human
history, details available on request.

In this first graph, I am showing the doubling time on the Y axis, with the
date on the horizontal.  The signal is chaotic at first, because there was
no real systematic search for enormous prime numbers.  It was almost as if
no one cared.  Shameful.  Unacceptable behavior from our grandparent's
generation.

I have applied a homemade smoothing and filtering technique I actually
designed for another purpose, but the point is that as you go down into the
higher series numbers, the curve gets smoother.  Check it out:

Very well, OK let's ignore those early years before 1978, when the earth was
inhabited by cave men and cave women, dwelling in yurts.  Then we can go to
a more interesting scale.  I am intentionally avoiding going with semilog.
Now we get this:

So now we see what happened when computers started coming online in the 70s
and we started using them for fun stuff like the Lucas Lehmer algorithm.  We
had a cluster of Mersenne discoveries.  I first heard of the Lucas Lehmer in
high school, which was in 1978.

Let's zoom in once more:

Do ignore the oddball horizontal scale for now.  I wanted to get that 1987
data point in there.  What we are showing here is that with successively
more data used in the smoothing technique, it is ever more clear that
something dramatic was causing the time between doubling size of the largest
known prime to be reduced dramatically.  So now if we look at another order
of magnitude closer, we see the start of the graph right around when GIMPS
was started with the first Prime95 FFT algorithm, which is around when I got
involved in it:

Clearly, the introduction of GIMPS with the optimized Prime95 has
dramatically lowered the time it takes to find cool new stuff!

This is why I commented earlier that this isn't exactly a record, but
depending on how you filter the data, it is a record.  We had an anomalous
interval there in the unfiltered noisy data, but look at that gorgeous
series 12!  Oh beautiful is that data!  If we take into account the trend in
the last dozen Mersenne discoveries, we see we are getting steadily faster.
You see there that weird cluster in the 2004-2006 timeframe that had blown
my mind.  I had expected a stretch like the last four years for a long time.

Friends, my apologies.  I just looked over what I have written and realize
that it must be incomprehensible to those who do not follow this kind of
Prime group; we take so much for granted.  But I realize that here I am
among a number of practically normal people in the area of obscure number
theory.  Perhaps many here are prime virgins.

spike

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