Known factors distribution graphs

[James Heinrich]

I've now got some graphs depicting the distribution of both quantity and size of known factors, as well as TF level for exponents with no known factors:

up to 192M: http://mersenne.ca/graphs/factor_bits_100M/
up to 4294M: http://mersenne.ca/graphs/factor_bits/

The smaller graph is focused on the most active range, and each column represents 100k range of exponents. The larger graph covers the entire range up to M2³² in my database, and each column represents 1M exponent range.

Gray background is total exponents in that range (the part you can see represents exponents with no known factor). Coloured parts of the graph represent known factors: overall height of the coloured portion (relative to the gray portion) is the number of exponents with known factors. Colours vary according to the distribution of size of known factors (per the legend), scaled such that the overall height of the coloured portion matches the total number of factored exponents. This can get a little weird if more than one factor is known per exponent, especially across larger ranges, for example between 122M and 133M someone has spent considerable time making sure everything is TF'd to the full level, not stopping after finding the first factor as is normally done (at least with Prime95).

Light red line represents default TF level that Prime95 would take the exponent to; light blue line is this plus 3 bit levels, approximating what level GPU72 would TF the exponent to. The black line represents average TF level for exponents in the range with no known factor.

[chalsall]

Originally Posted by James Heinrich

I've now got some graphs depicting the distribution of both quantity and size of known factors, as well as TF level for exponents with no known factors.

VERY COOL!!!

It would be interesting to know just how many GHz Millennium this knowledge represents!

[Dubslow]

Originally Posted by chalsall

VERY COOL!!!

Ditto!

Great job, James!!!!

[James Heinrich]

Originally Posted by chalsall

It would be interesting to know just how many GHz Millennium this knowledge represents!

30.

Well, as close as I can approximate it, anyways.

Going on a few assumptions:

• only TF was done (ignoring P-1 and ECM effort, ignoring TF beyond the first factor, ignoring LL, etc)
• exponents with no factor were all TF'd to the average level for that 1M range
• exponents with factor would have found that factor by TF exactly halfway between the minimum TF bitlevel (k=1) for that range and the average no-factor TF level for that range [this is the big important assumption, someone correct me if I'm wrong]. Of course, half the bitlevel means very much less than half the effort, which is true: the vast majority of the known factors are cheap to find (I found 40 million last week on a single CPU, for example).

With the above assumptions, I get a total of 11,087,435 GHz-days effort, which works out to just over 30 GHz-millennia.

[chalsall]

Originally Posted by James Heinrich

With the above assumptions, I get a total of 11,087,435 GHz-days effort, which works out to just over 30 GHz-millennia.

Wow!!! Thanks for that.

I wonder how many dinosaurs died to provide the kWhs needed....

[cheesehead]

All of them. as well as a bunch of banana peels in my local landfill^*. :-)

- - -

^* Landfill is tapped to gather methane gas which is then used to generate electricity.

[James Heinrich]

Around here: none. All our power comes from hydroelectric, wind and solar.

[chalsall]

Originally Posted by James Heinrich

Around here: none. All our power comes from hydroelectric, wind and solar.

Cool.

Here in Barbados we're not so smart... we burn diesel for all of our electricity. Which is pretty stupid considering we get approximately 12 hours of sunshine year round. Plus, a heavy draw is air conditioning, which is mostly used during the day....

[James Heinrich]

It's only in the remote snowy places that we use solar. So that you have to go out and shovel the panel at 10am sunrise to get power to your block heater
In densely populated places, naturally, we use better things. Like nuclear.

[VictordeHolland]

What is causing the "weird" lines between 20M-40M and 120M-135M ? Were these parts poorly TF-ed to 2^50 or something?

[James Heinrich]

Originally Posted by VictordeHolland

What is causing the "weird" lines between 20M-40M and 120M-135M ? Were these parts poorly TF-ed to 2^50 or something?

I'm not entirely sure about the cause, but the graph shows an abnormal shift towards more larger factors in those ranges. The height of the graph bar shows the number of factored exponents in that range (compared to the number of candidates represented by the grey behind it). The colour distribution represents the distribution of factor bit sizes.

If I had to offer a theory, I'd say that most of the PrimeNet range has had factoring done and stopped upon finding the first factor, but in those ranges I'd speculate that someone has done TF to (2^66?) but not stopped after finding the first smaller factor. So you end up with proportionately more factors in the 2^55-2^66 range that wouldn't have been found if TF was stopped as soon as the first factor was found. That's my theory anyways, competing theories are welcome.