option for finding multiple factors during trial factoring

[tha]

Originally Posted by R.D. Silverman

What purpose is served by finding more than one factor?

On the mersenne.ca site there is a a graphic showing the factors found by the clients. In the approx. 121M to 132M range there is a sudden discontinuity that, if the graphic is correct, would indicate a noticable lack of factors between rougly 24 bits and 56 bits. There are some other apparent smaller discontinuities in lower ranges.

I am testing some theories; maybe a client was used with different settings than all or most other participants, or a faulty machine was used to do some ranges. I redid 2000 exponents in half an hour on a single core in the 129M range and I am comparing it to the GIMPS database. I am still working on that in a spreadsheet.

[TheMawn]

Originally Posted by R.D. Silverman

What purpose is served by finding more than one factor?

Because he f-----g feels like it.

[R.D. Silverman]

Originally Posted by tha

On the mersenne.ca site there is a a graphic showing the factors found by the clients. In the approx. 121M to 132M range there is a sudden discontinuity that, if the graphic is correct, would indicate a noticable lack of factors between rougly 24 bits and 56 bits. There are some other apparent smaller discontinuities in lower ranges.

I am testing some theories; maybe a client was used with different settings than all or most other participants, or a faulty machine was used to do some ranges. I redid 2000 exponents in half an hour on a single core in the 129M range and I am comparing it to the GIMPS database. I am still working on that in a spreadsheet.

Aha! Tracking down a potential problem! This has value.

[Jayder]

Originally Posted by R.D. Silverman

Aha! Tracking down a potential problem! This has value.

Do you have a list of officially sanctioned projects somewhere? Speaking as a total noob, I would love to know what things do and don't have value, as I would hate to Waste Time™.

[R.D. Silverman]

Originally Posted by Jayder

Do you have a list of officially sanctioned projects somewhere? Speaking as a total noob, I would love to know what things do and don't have value, as I would hate to Waste Time™.

Project value is defined by opinions of professional mathematicians.
Such value judgments are made all the time as we are asked to review
project applications submitted to the government for funding (e.g. NSF).
We are also asked to make such judgments every time we referee a
paper.

If you want to know what things have value (to the mathematical community), I suggest you start by getting
a PhD in mathematics. This will teach you what are the current topics of interest in mathematics. Read the literature. Read published papers.

If you were to take a poll among professional number theorists you would
find very little interest in finding isolated prime cofactors of Mersenne
numbers. They are mere numerical curiosities.

Of course, your time is always yours to waste as you desire.

[VictordeHolland]

I've looked at the graph many times and wondered the same thing. It has been discussed before in the following thread:
http://mersenneforum.org/showthread.php?t=17177&page=2

It might be due to candidates with multiple factors known. Of those candidates not the smallest factor is taken when creating the graph. Take for instance M124000111 and M124000561, both have 3 factors known. In the first one the biggest factor known is listed first on mersenne.ca , while in the second the smallest factor known:
http://www.mersenne.ca/exponent.php?...0111;124000561

[TheMawn]

Originally Posted by R.D. Silverman

I suggest you start by getting
a PhD in mathematics.

Of course, your time is always yours to waste as you desire.

Then forgive me if I don't get a PhD in mathematics.

[R.D. Silverman]

Originally Posted by TheMawn

Then forgive me if I don't get a PhD in mathematics.

The matter under discussion was how to recognize what math projects
have value. This requires advanced training and experience in mathematics.

You are free to waste time any way you want.

[Jayder]

I am genuinely curious in reading about what projects have value to the mathematical community and why. So far I have gathered that Home Primes are rubbish, and Cunningham numbers are fine. If the only way for me to read about these things is by getting a PhD, then I guess I will never learn them. Thank you for providing a bit of insight, anyway.

[R.D. Silverman]

Originally Posted by Jayder

I am genuinely curious in reading about what projects have value to the mathematical community and why. So far I have gathered that Home Primes are rubbish, and Cunningham numbers are fine. If the only way for me to read about these things is by getting a PhD, then I guess I will never learn them. Thank you for providing a bit of insight, anyway.

How much abstract algebra/group theory do you know?
Do you understand that factoring (prime base) Cunninghams
reveals the structure of finite fields? That factorizations of 2^n-1
are tied historically to perfect numbers, that the infinitude of Mersenne
primes is an open question? Indeed, we don't even have a proof
that 2^p-1 is composite i.o. for prime p.

There are many "connections" between Cunningham numbers
and problems in number theory. Try reading R. Guy's book.

Mathematicians like to look at general settings for problems.
Interest in factoring Fibonacci/Lucas numbers came from generalizing the
Cunninghams to other linear recurrent sequences.

Trying to find a prime cofactor of a very large Mersenne number
isn't very interesting because they are ISOLATED CURIOSITIES.
An attempt to factor ALL M_p for (say) p < 10^8 is so far out
of computer range that it isn't under consideration. Finding
small factors of M_p speeds the search for Mersenne primes by
eliminating many numbers from having a full LL test (which is
expensive). A *systemic* investigation for ALL M_p to find
PRIME cofactors would be harder than GIMPS itself.

The search for Mersenne primes continues because it is FUN.
They also provide an incentive to develop new code and test
innovations in computer hardware (e.g. GPUS)
There is almost no other justification for it. Finding another
Mersenne prime would not help us prove any theorems.

If you really want to find (say) 10M digit primes it is far easier to
generate them via Maurer's algorithm or using my improvement to
Maurer's method. [Maurer's method doubles the size of the
primes at each iteration; mine triples the size]

It's not that Home Primes are rubbish. But they are UNCONNECTED
to other parts of mathematics. They are just a recreational hobby.

The factorizations that arise in the Cunningham project have little
value. But they do make a handy BENCHMARK for working on and
improving factoring algorithms. And they have historical interest.

What makes something have value in math is mostly determined by
a couple of things:

(1) Is it connected to something else? Can the result be used elsewhere?
(2) Does seeking the result yield new insight, ideas, techniques, algorithms, theorems?

[Jayder]

Thank you for your response. I will do some reading on Maurer's algorithm, and I'll take a look at Richard Guy's book. I hope I will be able to understand them well enough.