[QUOTE=chalsall;268697]Bob... With all due (and great respect), you must agree that many great scientific breakthroughs have occurred because the observed empirical evidence doesn't agree with the mathematical models' predictions.

Newton -> Einstein -> Bohr et al is just one example....[/QUOTE]

That ain't quite our expectations here, but it's a nice thought
all the same:smile:

Cheers!

David

[QUOTE=R.D. Silverman;268675]You do not know if you have the right answer[/QUOTE]

But I'm not confident that have the right answer if I do a complicated analytic approach either, without looking at the data - it seems unlikely that this is a situation where we can get away without numerical integration, possibly in several dimensions - and it's much more likely that I've dropped a factor of zeta(2) somewhere in an integral, or a factor of two because I hadn't considered some higher reciprocity law restricting the possible prime factors, than that I'm facing an absurd statistical anomaly in the large volume of already-gathered data.

[QUOTE]
I recall a comment from a class that I took from W. Zangwill in grad school.
We were discussing an O.R. problem and Prof. Zangwill asked a generic
question about how to go about solving the problem under discussion.
The first suggestion was "go out and gather statistics and look for a pattern'.
Prof. Zangwill's response was 'this is a truly horrible way to do it; may we
have a better suggestion'. I then spoke up: "Start by defining the variables
that will affect your decision". Zangwill's reply was that this was
precisely the correct way to approach an optimization problem.

Perhaps you think you know more than Willard?
[/QUOTE]

I think that it's worth figuring out in advance the amount of effort you want to devote to a problem, and going out and gathering some stats may well tell you that the problem can be solved without difficulty and it's not worth the bother of defining non-trivial variables. Particularly if other people have already gathered the stats very comprehensively for some other purpose.

[QUOTE]This is NOT a 'complicated model'. It is very straightforward. The only
hard part is (as I have said) working out the details of how to compute the
conditional probability of P-1 succeeding after TF has failed.[/QUOTE]

So show us how to do that; is there more to it than the integral from the TF limit to infinity of the probability that the candidate has a factor of that size times the Dickman-[tex]\rho_2[/tex]-function-based probability that a factor of the appropriate size can be found with the given B1/B2 bounds?

This seems like a problem which is entirely a matter of feeding the right functions to your numerical integrator in the right order - I will be amazed and delighted if there are any nice cancellations so you don't have to do the numerical integrations.

[QUOTE=davieddy;268700]That ain't quite our expectations here, but it's a nice thought all the same:smile:[/QUOTE]

My point stands.

Mathematics is nothing but a model when dealing with most of the Real World.

Models can be wrong; and thus their predictions should always (and continuously) be compared to empirical evidence before they are implicitly trusted....

(Amusingly (interestingly), my post quoted (in its entirety) by daveddy was deleted. Hmmm....)

I know we are not in the soapbox yet but...
 

I think a couple of posts may have got deleted.
I'll check again, but I can remember them verbatim anyway.

I do hope the OP is content with his/her answer to the (seemingly)
innocuous query.

David

Causality is bunk
 

[QUOTE=chalsall;268703]My point stands.

Mathematics is nothing but a model when dealing with most of the Real World.

Models can be wrong; and thus their predictions should always (and continuously) be compared to empirical evidence before they are implicitly trusted....

(Amusingly (interestingly), my post quoted (in its entirety) by daveddy was deleted. Hmmm....)[/QUOTE]

Dr Who just deleted a few posts and killed my father before I was born:(

David

[QUOTE=R.D. Silverman;268693]One hopes that these people are not involved in 'designing bridges' or other
similar activity.

Another hackneyed phrase: "they don't set the bar very high, do they?"[/QUOTE]

I'm glad Isambard Kingdom Brunel wasn't you.
Hackney wasn't a good place to be today.
Jessica Ennis could outjump you anyday.
Alternatively, see me with a pair of stilts.
Or were you Limbo dancing?

Put that in your pipe and smoke it.
(But make sure you don't inhale!)

David

[QUOTE=davieddy;268709]I'm glad Isambard Kingdom Brunel wasn't you.[/QUOTE]

There are persons who are known for thinking...

There are persons who are known for doing...

There are persons who are known for thinking [U][I]and[/I][/U] doing...

Thank your god(s) for all.... :smile:

[QUOTE=R.D. Silverman;268693]One hopes that these people are not involved in 'designing bridges' or other
similar activity.

Another hackneyed phrase: "they don't set the bar very high, do they?"[/QUOTE]

Bob:

I think you'd be absolutely amazed at the "Good enough" used in designing famous bridges such as the one in Brooklyn. Structural engineers refer to a "factor of ignorance", meaning they don't know the EXACT maximum loads, or the EXACT behavior of the materials involved close to their maximum loads.

.....

*******
So, the quick heuristic: GIMPS is trying to
a) prove that the higher known mersenne primes have no other mersenne primes between them
b) find new mersenne primes.
While minimizing the effort involved and the patience required.

Since almost all mersenne numbers are not prime, and there are very many mersenne exponents to test, a very small error is introduced by counting only the expected effort to show that a given mersenne number is not prime. This can be accomplished by
a) Showing the mersenne exponent is composite (trivial, done at the server side)
b) TF effort finding a factor
c) P-1 effort finding a factor
d) Two LL tests finding a nonzero residue
e) P+1 effort finding a factor
f) ECM effort finding a factor

We will assume that the compute resources are fixed, and we will count one CPU running for one day the same effort as one GPU running for one day. [yes, very attackable -- but makes the math more tractible -- and the match in purchase price and operating costs are both reasonable]

One GPU can do about 128 times as much TF as one CPU for the same effort. (Empirical fact; applies to high-end GPUs versus recent CPUs). One GPU can do about 4 times as much LL for the same effort as a CPU. At the moment, GPUs do not do P-1, P+1, or ECM. Current GPU TF also uses a significant amount of CPU power. [Programming resources are required to change this, and they are significantly more scarce than computing power]

By a well-known theorem, we are at an optimum when, on average, it doesn't matter which method we choose to determine that an average mersenne number is not prime.

Assuming the original CPU settings from the server for P-1 and TF were at that optimum, and that, each bit level of TF has a 1/70 chance of finding a factor (bit levels are close to 70), but each additional bit level costs twice what the previous one did. If, for the previous CPU I didn't care at a given bit level, and I can get 128 times as much work out of my GPU on TF now, I should do log2(128) additional bit levels = 7 to reach the same tradeoff.

This ignores, of course, that my GPU can do LLs, too, 4x faster, so I should only do log2(128) - log2(4) = 5 additional bit levels with my GPU. As the GPU is so much more effective at TF than my CPU, it makes no sense to do TF on my CPU at all.

It also ignores the relative scarcity or abundance of GPUs with respect to CPUs. There are no CPU-less systems with GPUs, although there may be a some systems with two GPUs in them, maybe a few somewhere with three or more. If there are just a few GPUs, then they may not be able to do all those TF bit levels. And if there are a lot of them, it may be much more effective to have most of them doing LL tests, although P95 tells me right now that there's a lot more TF than LL happening on GPUs for GIMPS.

Finally, it occurs to me that I am working on mfaktc right now more because I think it is important for me to finish my commitments than I do because I think this behavior is actually optimum in terms of finding the next mersenne prime. If we do 7 additional bit levels of TF at 70 bits, that's approximately 10% of the remaining mersenne numbers eliminated. It still leaves the other 90% to be LL tested and double-checked. A 4x speedup in the LL test is *much* more important to reducing the total effort required.

I think this relatively low probability of non LL tests finding factors with reasonable amounts of TF, or P-1 effort is what leads to the relative broadness of the maximum.

[QUOTE=chalsall;268703]My point stands.

Mathematics is nothing but a model when dealing with most of the Real World.
[/QUOTE]

Since when is an algorithm for finding Mersenne primes "part of the real
world".?? None of what we are doing is remotely related to modelling the
real world.

Models can be wrong. The one I presented for the expected time to
test a Mersenne candidate for primality is not. [I never actually wrote down
the objective function, but it follows easily from the definitions that I
gave] If you believe that it is, please tell us what is wrong.

[QUOTE=R.D. Silverman;268722]Models can be wrong. The one I presented for the expected time to test a Mersenne candidate for primality is not.[/QUOTE]

Ah, so then you agree before this Court of Public Opinion that Models can be wrong.

[QUOTE=R.D. Silverman;268722]Since when is an algorithm for finding Mersenne primes "part of the real
world".?? [/QUOTE]
Let's see...real people, real money for prizes, real power from real lumps of coal to run the real machines....