Theoretical Model

[aymanmikhail]

It seems the problem of finding larger and larger primes is that computer power is the limiting factor. Is it not possible to develop a theoretical model to find and verify larger and larger primes without extensive computation?

[Christenson]

For any theoretical model you care to create (and we'd love to have new ones), the problem is that the amount of computation involved is going to be enormous, because the numbers themselves are enormous. And, even if you did find a model that caused the work to get 10 or 100 times as small, we computational number theorists would simply go out and find larger numbers to work on....

Now, if you want to talk about what that theory might be, I'd point out that the LL test is pretty straightforward. MPQS isn't too bad, but there's only a few here that understand SNFS or GNFS. But I will warn you that you need to do some serious studying, and be smart, even to understand the existing tools. I think some may be game, but I found helping the programming effort to be a lot easier, and even that is slow work for me.

[jasong]

Another problem for anyone trying to improve the theory is that they decide to present their theory on Mersenne Forum first, which is likely to illicit flames from some of the better educated/unrefined users. I happen to keep in touch with someone who's actively trying to improve the art of number theory in general, and while he might search the forum for help in various things he's working on, he hasn't actually signed in in a very long time.

As it's been said before, while a lesser man may make a lot of mistakes, a greater man will make even more mistakes simply because he continues to try.

[Uncwilly]

Originally Posted by aymanmikhail

It seems the problem of finding larger and larger primes is that computer power is the limiting factor.

Do you have a true understanding of the size of the numbers we are talking about? They have no real connection to the numbers found in the physical universe. 13,000,000 digits long. The physical universe typically needs less than 100 digits to describe it.

[Christenson]

Originally Posted by jasong

Another problem for anyone trying to improve the theory is that they decide to present their theory on Mersenne Forum first, which is likely to illicit flames from some of the better educated/unrefined users. I happen to keep in touch with someone who's actively trying to improve the art of number theory in general, and while he might search the forum for help in various things he's working on, he hasn't actually signed in in a very long time.

As it's been said before, while a lesser man may make a lot of mistakes, a greater man will make even more mistakes simply because he continues to try.

Indeed, it's time to remind that certain flamer to tell us why he said a certain paper, after corrections, was still unpublishable....remind me to get in touch when I start studying number theory again.

http://mersenneforum.org/showthread.php?t=13600

Don't let him get to you.

[cheesehead]

Originally Posted by aymanmikhail

Is it not possible to develop a theoretical model to find and verify larger and larger primes without extensive computation?

We'd welcome such a model!!

No one has yet come up with one that's more streamlined (asymptotically faster) than Lucas-Lehmer, but there's no proof that a better digital algorithm is impossible. (There is wide skepticism that one is possible, yes, but not proof of impossibility.)

Of course, many think that quantum computing should eventually be able to find and verify large primes faster than digital computing can.

[davieddy]

Originally Posted by cheesehead

Of course, many think that quantum computing should eventually be able to find and verify large primes faster than digital computing can.

If by "many" you mean UncWilly and Chalsall, I will endorse that

But then again, as Feynman, Bohr, Einstein, Born, Borg...
have said before:
If you think you understand QM, you don't.

Or was that the 60s?
I've forgotten.

David

[aymanmikhail]

No; I do not have such a model. I apologize for the implication by stating the obvious.


Yes; we are dealing with infinity. I would argue that these numbers do exist in the physical universe just as gravitational singularity at the center of a black hole exists.


Within the current state of human knowledge/technology, the persons with the most computing power will find the next largest prime. The human mind is slow but it possesses imagination/creativity and the ability to think about the infinite.

[jasong]

Originally Posted by davieddy

If by "many" you mean UncWilly and Chalsall, I will endorse that

The fun thing about quantum computing is that they're managing to add a qubit to what's already working about once every year or 2. So, basically, qubits are advancing at about the rate of Moore's Law, which means they'll be pretty much useless for another 30 years or so. Of course, things might begin to speed up. But, for the moment they're like the science of fusion, it's a future technology, and will be a future technology for a long time.

[xilman]

Originally Posted by aymanmikhail

Yes; we are dealing with infinity. I would argue that these numbers do exist in the physical universe just as gravitational singularity at the center of a black hole exists.

Now that is an interesting statement.

Could you give us some pointers to any literature which contains any experimental or observational evidence to support your claim about the existence of singularities please?


Paul

[aymanmikhail]

Originally Posted by xilman

Now that is an interesting statement.

Could you give us some pointers to any literature which contains any experimental or observational evidence to support your claim about the existence of singularities please?

Paul

Mathematics is a natural science of which infinity is not a part of?

The Penrose–Hawking singularity theorems are a set of results in general relativity which attempt to answer the question of when gravitation produces singularities.
http://www.ias.ac.in/jarch/jaa/17/213-231.pdf