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Old 2013-10-06, 16:08   #1
Tony Reix
 
Oct 2002

418 Posts
Default 500€ Reward for a proof for the Wagstaff primality test conjecture

Hi,

I've decided to offer a 500€ reward for the first guy/lady who will be able to provide a proof for the conjecture that Anton Vrba and myself have found, years ago, about proving the primality of a Wagstaff number.

I've just started the process of defining the rules and putting in place some pages on the Web. I have much more work to do. So, the first version is very raw: http://tony.reix.free.fr/WagstaffReward/ . Anyway, I plan to improve these pages. I have to refresh my memory about HTML, Lyx, and LaTeX stuff.

Such a candidate proof will have to be reviewed by peers and to be published in a well-known Mathematics journal.

I also have to take decisions about if proving one of the two other conjectures (Mersenne and Fermat) should deserve a reward.

I also need approval from Anton Vrba, and provide thanks to other people who provided help to this.

I am also looking for help and support. Anyone who would like to read candidate proofs, or anyone who would like to sponsor the reward (give money !), is welcome.
One important help would come from Number Theory experts to give their opinion about the importance of these conjectures and to write it down.

In my opinion, it is worth to give light on these conjectures due to several reasons: 1) Number Theory books are wrong ! Yes, LLT test is not only for Mersenne numbers and N+1 world is not for LLT and N-1 world for other stuff. As an example, 4 years after me, someone published a proof that LLT test for Fermat numbers is equivalent to Pépin's test. 2) Having a primality test for Wagstaff numbers will extend the GIMPS search and it will speed up finding a 100 millions digits prime, because there are more Wagstaff primes than Mersenne primes; 3) it's fun ! 4) There is a powerful and magic Math world in these DiGraph cycles.

I'll work on this during the next months since, in November, I'll be in Asia (Singapour, Bali, Angkor, Luang Prabang) for fun and photography, and I will not be able to do anything. Moreover, I'm now involved in preparing my trip, plus many other stuff dealing with my photographer activity.

My goals are: 1) get famous ! (more seriously, I would be VERY happy that all the time I spent working on this damned and hard Math stuff would lead to something real and useful) and thus appear in Number Theory books, with Anton Vrba and the prover of the conjecture; 2) let my children be proud of their damned father; 3) bring attention to Math theory more than to prime records; 4) extend our knowledge about this area and bring Mathematicians to explore this LLT magic world !

Also, I know that Paul Erdős would have liked this reward ! Though my Erdős number is probably close to infinity !
More seriously, I DO believe that studying the properties of the Cycles of a Digraph under S^2-2 \ mod(N) will bring very powerful theorems. And, if you would read Lucas papers, as I did, you would know that he found a lot of things (but without valid proofs ! ).

Any comment or suggestions are welcome !

Regards
Tony Reix
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Old 2013-10-06, 19:29   #2
Batalov
 
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Mar 2008
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Tony, you may want to post to NMBRTHRY list.

It is a slow list (posts are approved once a week or so), but there are many people there who will pass their professional opinion. (The intersection of that list with this forum is in single digits, ever wondered why?)

When you will post there, you will do better by dropping the second half that starts with "In my opinion, it is worth to give light on these conjectures due to several reasons: 1) Number Theory books are wrong" etc. Firstly, this bumps the crank score so that you will lose half of your audience right away. Secondly, all of it is already in the conjecture. Post it in plain text without any fancy attachments.

A 500 Eur Conjecture:
Let: N_q = 2^q + 1\ and\ W_q = N_q / 3 (a Wagstaff number), and S_0 = 1/4 \ (mod\ N_q),\ S_{i+1} = {S_i}^2-2\ (mod\ N_q) for 0<i<q.
Prove W_q is a prime iff S_{q-1}\eq S_0 (mod\ W_q) and there is no 0<i<q-1 such that S_i \eq S_0 (mod\ N_q)

or on that list, they imply TeX between $ signs:

Code:
A 500 Eur Conjecture:
Let: $N_q = 2^q + 1 and W_q = N_q / 3$ (a Wagstaff number), and $S_0 = 1/4  (mod N_q), S_{i+1} = {S_i}^2-2 (mod N_q)$ for 0<i<q.
Prove W_q is a prime iff $S_{q-1} \eq S_0 (mod W_q)$ and there is no 0<i<q-1 such that $S_i \eq S_0 (mod N_q)$

More details at ...<leave your link>
This (above) is basically all you need to post.
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Old 2013-10-06, 20:48   #3
xilman
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Quote:
Originally Posted by Batalov View Post
NMBRTHRY"] (The intersection of that list with this forum is in single digits, ever wondered why?).
Are you sure of that? You may be correct but I already know of you, me, Bob, Bruce, Tom W, Phil Carmody and there are probably a few others I forget right now. Perhaps other NMBRTHRY lurkers could make themselves known.
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Old 2013-10-06, 21:15   #4
Batalov
 
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To be honest, I didn't count myself. I am a lurker there.

I agree that the count would be somewhat higher if you included people who essentially left this forum, like, e.g. maxal.
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Old 2013-10-06, 22:06   #5
only_human
 
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Sep 2002

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I lurk there. I've given up thinking about Zhi-Wei Sun's conjectures though. Before I can think about one, it seems there are a dozen more.
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Old 2013-10-07, 00:48   #6
CRGreathouse
 
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Quote:
Originally Posted by xilman View Post
Are you sure of that? You may be correct but I already know of you, me, Bob, Bruce, Tom W, Phil Carmody and there are probably a few others I forget right now. Perhaps other NMBRTHRY lurkers could make themselves known.
I'm one.
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Old 2013-10-09, 19:34   #7
Tony Reix
 
Oct 2002

418 Posts
Default Review of the Conjectures

I've asked Hugh Williams his opinion about the conjectures. He accepted and he first wanted to look at the conjectures and the proofs of necessity. He is not really sure that they are correct or provable.
So, I"ll wait for him to give me his opinion and recommendations.
It is very kind of him to take time looking at this. However, he spent so many hours, days, weeks, months, and years on this subject that his opinion does count.
Wait & See.
If these conjectures are nuts, I'll be free to spend more time on taking photographs ! ;) and I'll buy a new Nikon lens instead !
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Old 2013-10-10, 01:23   #8
maxal
 
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It may be worth to point out the previous discussion on this topic inspired by Anton Vrba's proposed proof: http://www.mersenneforum.org/showthread.php?t=10737
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