GIMPS' Prize.

[T.Rex]

Hello,

Though we have often many interesting discussions on GIMPS' Math forum and though the Wiki does help to gather properties and information about Mersenne numbers and LLT, very few really serious and new does appear. And I do not see on the web many new papers studying Mersenne numbers, nor papers proving new interesting properties.

Though the EFF prize could be partly awarded to the discoverer of a Mathematical breakthrough in searching for Mersenne primes, I never saw any (serious) attempt.

Though there are many interesting and useful papers in forgotten old dusty books, few of us can get access to and read them.

So, I propose we organize a "GIMPS' prize" for encouraging searchers and students in Number Theory to write papers about Mersenne numbers and about the LLT:
- the papers could either a) gather old forgotten properties with proofs, or b) explain in details complex algorithms (FFT, ...), or c) provide new studies about properties around Mersenne numbers of LLT. With a priority to: c), a) then b).
- the prize should be modest (500 to 1000 $US or Euros) and the money could be provided by the GIMPS contributors (I'm OK to give about 10 Euros. If we all give a dozen of $US/Euros by means of Paypal, it will be easy to collect the value of the Prize)
- the prize would be given once a year to the best proposed paper and the decision should be made by a board including George and many of the Mathematicians that contribute to the GIMPS.
- all interesting papers should be published on GIMPS' web-site as a (more detailed) contribution to our GIMPS Wiki which is aimed to provide short and simple explanations to beginnners.

I think this would encourage more students and professors to have again a look at these (old but still not perfectly understood) Mersenne/LLT areas.

Please provide your comments and say if you would agree to contribute with a dozen of Dollars/Euros to build the prize.

Regards,

Tony

[ewmayer]

If one discovers new mathematical properties of the kind you mention and they are truly nontrivial/interesting, publication in a peer-reviewed journal would seem like the best kind of reward, would it not?

[T.Rex]

Originally Posted by ewmayer

If one discovers new mathematical properties of the kind you mention and they are truly nontrivial/interesting, publication in a peer-reviewed journal would seem like the best kind of reward, would it not?

Yes, sure. But I do not know about a Journal specialized in the Mersenne/LLT area.
There is the Fibonacci Quarterly. It sometimes talks about subjects close, like Pell' numbers and Lucas Sequences, used for proving kinds of LLT, but its main subject is not really about Mersennes and LLT. And you have to pay in order to read the papers ...
So, I'm not aware about a Journal (paper or web) that can attract/encourage people to study the Mersenne/LLT area. Include the study of Fermat numbers (primality can be proved by LLT) and LLR (for k.2^n-1 numbers) could make such a GIMPS' prize more interesting.
(Some few) money and fame could help to have more people interested by the GIMPS' project AFTER the 10Million prime is found. We have to think about a strategy for AFTER this event.

Remember when I found the property: . It does not help proving primality nor factorizing Mersenne composites. But it is useful because it increases our knowledge about Mersennes. Maybe one day someone will be able to link together several small properties found/published thanks to such a "GIMPS' prize" and then publish a paper showing a real progress (primality/compositeness proof, or factorization) in a peer-reviewed journal.
About the DiGraph under x^2-2 modulo a Mersenne prime, it would be nice to have students interested to continue the study.

Also, we could imagine the "GIMPS' prize" to reward a paper already published in a peer-reviewed journal.

So, the main idea is: Find a way to attract more Number Theorists to study again Mersenne numbers and LLT. They may find a result which can improve the performance of the GIMPS project and this research activity can attract more contributors.
The question is: Is a "GIMPS' prize" the appropriate solution ?

George, any comments ?

Tony

[m_f_h]

Originally Posted by T.Rex

(...)money and fame could help to have more people interested by the GIMPS' project AFTER the 10Million prime is found.

I like your idea of a "pool" of many people working on something and everybody giving a small amount of money to constitute a prize -
in some sense a kind of "lottery" where tickets are mathematical results...
(maybe each member of the community having the possibility to vote for approval or rejection when (say: each time) a winner is to be dedicated.

Originally Posted by T.Rex

Remember when I found the property: .

Is there a reference about you finding that? It looks quite elementary (writing n=((n+1)/2)²-((n-1)/2)² as for any odd integer n, then 8x = 2^(q-1) = 1 mod 3, mod q for any odd prime q>3) so it would astonish me if no mathematician had ever noticed that before. Is there some use in writing Mq like this?
PS: sorry for my ignorance and if I don't capture well the meaning of x...

[T.Rex]

Originally Posted by m_f_h

Is there a reference about you finding that? It looks quite elementary (writing n=((n+1)/2)²-((n-1)/2)² as for any odd integer n, then 8x = 2^(q-1) = 1 mod 3, mod q for any odd prime q>3) so it would astonish me if no mathematician had ever noticed that before. Is there some use in writing Mq like this?
PS: sorry for my ignorance and if I don't capture well the meaning of x...

Look at this proof. It is a little more complex than you say. About someone who may have found the property before me, that's possible ; but I never found it in any book or paper I've read. Up to now, it seems this property (which is the consequence of basic features of Mersenne numbers) is not very useful ... , but it enables to see the number differently: look at this figure for q=11.
Tony

[m_f_h]

Originally Posted by T.Rex

(...) It is a little more complex than you say.(...)

I'm sorry if I don't understand why it is more complex than I say:
[ (n+1)/2 ]² - [ (n-1)/2 ]² = n, for n = Mq = 2^q-1, gives : [ 2^q/2 ]² - [(2^q-2)/2 ]² = Mq
The first term is the square of 2^q/2 = 2^(q-1) = 8x with x=2^(q-4)
The second term is the square of (2^q-2)/2 = 2^(q-1) - 1 = 3qy
since it is a multiple of q by Fermat's little theorem, and a multiple of 3 since if q is odd then 2^(q-1) is a power of 4 which is congruent to 1 mod 3, so with -1 this gives 0 (mod 3).

The only thing which is not trivial to me in that story is why you "mystify" the first term by writing it as 8x and not as 2^(q-1).

But I will study your picture which seems much less trivial to me...

[T.Rex]

Originally Posted by m_f_h

I'm sorry if I don't understand why it is more complex than I say...

Just read carefully the theorem and the proof.

If the Mersenne is prime, then there is only one way to write it as X^2-Y^2=Mq. It is the way you are talking about.
But, if the Mersenne is not prime, thus there are one or more other ways (depending on the number of factors of Mq) to write Mq=a*b=X^2-Y^2. In that case, X=8x and Y=3qy .
For each (a,b) such that Mq=a*b there exists (x,y) and (S,D) as explained in the theorem. And the proof makes use of stronger properties of Mersenne numbers.

As an example:
q=11 Mq= 1*2047 = 23*89 = 1024^2-1023^2 = (8*7)^2-(3*q*1)^2 = (1+5*q)^2 -(3*q)^2

Try with a higher q, like 109, which has 2 factors.

Do you understand now ?

Tony

[m_f_h]

Originally Posted by T.Rex

(...) Do you understand now ?

Thanks for your patient explanations - I indeed seemed to have a "reading problem" since I did not at all capture that it's about /different/ writings of Mq as (8x)²-(3qy)².
(I hope I understand this correctly now...)

[T.Rex]

Originally Posted by m_f_h

(I hope I understand this correctly now...)

I hope so.
As I said previously, I've found no usage for this new property. This property is another way to see the format of the factors of a Mersenne composite. I haven't found such a property for Fermat numbers.

Regards,
T.