Talk:Lucas–Lehmer primality test

Possible mistake in article

there is a mistake in the article , it supposed to :
$s_{p-1}\equiv 0{\pmod {M_{p}}};$
not :
$s_{p-2}\equiv 0{\pmod {M_{p}}};$

—The preceding unsigned comment was added by 85.250.124.207 (talk • contribs) 07:26, 4 November 2005 (UTC).

I disagree. $M_{3}$ is prime and also $s_{1}\equiv 0{\pmod {M_{3}}}$. That is, $14\equiv 0{\pmod {7}}$. However, $s_{2}\ \not \equiv \ 0{\pmod {M_{3}}}$; equivalently, $194\ \not \equiv \ 0{\pmod {7}}$. Therefore I think the article is correct as is stands. Eric119 03:35, 5 November 2005 (UTC)

$s_{0}=4$, so is $s_{p-2}\equiv 0{\pmod {M_{p}}}$ and not $s_{p-1}\equiv 0{\pmod {M_{p}}}$.

Dumbing this down with an example?

I think it might help some people by giving an example of how the Lucas-Lehmer test can prove primality of say... 2^5-1, for instance. As the article stands, not many people without prior experience of advanced mathematical notation and terms, would be able to take anything useful from this.

Great idea. Dcoetzee 03:24, 10 May 2007 (UTC)

-Yeah I'd like to see this in simple English. Sorry for not knowing wiki markup and doing this officially. — Preceding unsigned comment added by 96.32.150.222 (talk) 00:22, 11 November 2012 (UTC)

A bit confused about high speed mod

I'm a bit confused about the high speed mod operation. The article gives this example:

916 mod 2⁵−1 = 1110010100 mod 2⁵−1

= 11100 + 10100 mod 25−1 
= 110000 mod 25−1 
= 1 + 10000 mod 25−1 
= 10001 mod 25−1 
= 10001 
= 17. 

However if I try it with 14 mod 7 I get this: 14 mod 2³−1

= 1110 mod 23−1 
= 1 + 110 mod 23−1 
= 111 mod 23−1 
= 7. 

Now I know that 7 mod 7 is 0, but how does the ggiven algo generate a zero ? Thanks —Preceding unsigned comment added by 149.173.6.50 (talk) 15:25, 20 November 2007 (UTC)

That's a good observation, actually - it's possible for the result to be exactly 2ⁿ−1, in which case you have to change it to zero. If it were any larger, you'd reduce it by doing another iteration. Dcoetzee 01:23, 21 November 2007 (UTC)

I think, there are little problems whit this reasoning. First of all, if the modulo operation gives us 0 as a result, then this leads to the next: We arrive at 0, then we square it, it becomes 0 again, then we substract 2, then it is -2. Now, due to the [[1]] article, we can define the modulo of -2, so:

-2 = M_p-2 (mod M_p)

but this computation, namely, that we add M_p to the resutl to get a positive integer is much different from the algorithm described in this article.

Morover, this article states:

"since s × s will never exceed M² < 2^2p"

This statement eables, that the value after the squaring operation be M2^2p, and thus, the value before the squaring operation be M². So if the resutlt after the modulo operation is M_p, then we dont't need to convert it to 0.

So it would be needed to clarify this article on my opinion.

Cleaning up the "sufficient" part of the proof

Is there any reason why we write Zq(sqrt(3)) out explicitly, instead of just calling it an extension field? What's more, if sqrt(3) is in the field, there is no extension and we should just use Zq, whose multiplicative group has order q-1; otherwise, it's an extension field and its multiplicative group has order q^2-1. I suppose you could "extend" it, to get a group with order q(q-1), but it's not clearer that way, at least not to a mathematician. —Preceding unsigned comment added by 64.142.4.173 (talk) 18:15, 27 June 2009 (UTC)

Removed article text

I removed "They consider it valuable for finding very large primes because Mersenne numbers are considered somewhat more likely to be prime than randomly chosen odd integers of the same order of magnitude," because it's simply not true. There have been exactly 47 Mersenne primes, and the first 40 are known to have none in between - out of 1.3 million or so possibilities checked. Your average odd integer has a way higher probability of being prime, hence why we only use the Lucas-Lehmer test to look for the biggest ones, not for large amounts of primes to use for other purposes. —Preceding unsigned comment added by 68.147.216.149 (talk) 04:16, 10 December 2010 (UTC)

I agree it should be removed but don't agree with the stated reason. The chance of a random integer n being prime is 1/log(n). If n is odd then it doubles to 2/log(n). A Mersenne number with prime exponent has a better chance because 2^p-1 has no prime factors below 2p. If random odd integers of similar size as the Mersenne numbers were tested then the expected number of primes up to the 40th Mersenne prime would only have been 9. However, this is not the reason for GIMPS to test Mersenne numbers. PrimeHunter (talk) 14:06, 10 December 2010 (UTC)

Historic (first) use on M257: no prime

Lehmer and his wife Emma "bought a Monroe electric desk calculator on the instalment plan, but could only run it in the day as it blinked the lights in their house and the one next door. … which Lehmer spent some 700 hours in testing c1932"[2].

"In 1922, Monroe introduced the electric version of its Model K non-printing calculating machine. It had a large external driving motor." according to [3]

[4]: "932 D. H. Lehmer develops an improved version of Lucas’ test and shows that M257 is not prime, taking two hours a day for a year."

Finest source, worth looking at [5]: "Emma and Dick moved to Lehigh University in 1932, … The work by the Lehmers was funded by a Penrose Scholarship granted to Vandiver by the American Philosophical Society. Part of the money went to renting a 10-10-20 electric Monroe machine xnumber/pic_monroe_electr.htm at a cost of US$25 per month." – The pictures shows a somewhat smaller model K with just 8-8-16 digits. I can well imagine that the bigger model’s motor made the lights flicker at meager 110 Volt supply. – Fritz Jörn (talk) 08:02, 16 February 2013 (UTC)

Proof of correctness, Necessity

Hi, I think mine is simpler

$(1+{\sqrt {3}})^{2}=2(2+{\sqrt {3}})$ $2^{2^{q-1}}(2+{\sqrt {3}})^{2^{q-1}}=(1+{\sqrt {3}})^{M_{q}+1}\equiv (1+{\sqrt {3}})(1+{\sqrt {3}}^{M_{q}})\equiv (1+{\sqrt {3}})(1-{\sqrt {3}})=-2$ $2^{2^{q-1}}=2^{\frac {M_{q}+1}{2}}=2$ thus $(2+{\sqrt {3}})^{2^{q-1}}=w^{\frac {M_{q}+1}{2}}\equiv -1{\pmod {M_{q}}}$

I do not know who wrote this, but I concur that a proof using $1+{\sqrt {3}}$ is much simpler and easier to follow than the current proof using $6+2{\sqrt {3}}$. I wonder whether there is any source for it so that it can be put in the article.—Emil J. 08:07, 24 October 2024 (UTC)

Urgent Request for Intervention Regarding Baseless Accusations and Personal Attacks

Dear Wikipedia Administrators,

I am writing to request immediate intervention concerning the baseless and inappropriate accusations made against my recent contributions. As a specialist in number theory with a doctorate in the field, I am deeply offended by the unfounded claims questioning my relationship with the subject matter of Mersenne primes and the legitimacy of my publication.

The comment made in the public discussion suggests that I have no connection to the topic, which is entirely false. My career, research, and achievements revolve around number theory, including significant contributions to the study of Mersenne primes. This is well-documented through my work, including invited lectures at prestigious universities and my doctorate from an institution affiliated with the Isaac Newton Institute in Cambridge.

Moreover, the claim that the journal in which my work was published is a "predatory journal" is both defamatory and incorrect. The Arab Journal of Basic and Applied Sciences is a highly reputable journal governed by strict peer-review standards, including a rigorous two-reviewer process for my publication. The journal is also under the supervision of the British publishing house Taylor & Francis. Not only did I not pay any fees for publication, but the claim that my work lacks peer review is baseless.

The user’s accusations are not only harmful to me personally, but they also impede the dissemination of accurate mathematical knowledge to the public. I contributed the information to share an important development in the understanding of Mersenne primes, a topic that is vital to the broader mathematical community.

I urge Wikipedia to address this blatant misuse of its platform for personal attacks and to uphold the principles of integrity and fairness. Allowing such comments to persist harms the credibility of Wikipedia as a trusted source of knowledge. I kindly request that you remove the defamatory remarks and take appropriate action against the user responsible for this unfounded attack.

Thank you for your attention to this matter, and I look forward to your prompt response.

Sincerely, Dr. Moustafa Ibrahim Doctor of Number Theory, Researcher and Speaker on Mersenne Primes — Preceding unsigned comment added by ProfMoustafa (talk • contribs)

The section about "Recent developments" is a clear case of vandalism.

The whole section should be promptly removed as it was shamelessly plugged by that "author". The referred article is a paid publication in a predatory journal (fee-for-publishing, no peer review).

Inserted by this edit - https://wiki.riteme.site/w/index.php?title=Lucas%E2%80%93Lehmer_primality_test&diff=1250825439&oldid=1221898013 Serge Batalov (talk) 04:21, 16 October 2024 (UTC)

He keeps inserting his drivel.

I suggest locking the article.

The "ProfMoustafa"'s edit has no relation to the subject of this article. Serge Batalov (talk) 07:16, 16 October 2024 (UTC)

I would like to clarify that the referenced article underwent rigorous peer review by two independent referees before publication. I did not pay any fees for the publication, and the journal follows a transparent policy like many other reputable open-access journals. Attempts to suppress this important discovery only serve to limit the public's access to knowledge. This work represents a significant advancement in our understanding of Mersenne primes and should be openly acknowledged as a valid and valuable contribution to the field. ProfMoustafa (talk) 07:27, 16 October 2024 (UTC)

Dear Wikipedia Community,

It is concerning that individuals without a background in number theory, like Serge Batalov, are making uninformed claims about highly specialized topics like Mersenne primes. Given his declaration as a biologist, it is questionable how he is allowed to speak on this subject. His comments are detrimental to knowledge dissemination because he may not fully appreciate the complexity of Mersenne primes and their applications. I ask that Wikipedia take steps to prevent individuals who lack the relevant expertise from blocking accurate, peer-reviewed content.

Furthermore, I hope Wikipedia is committed to supporting diverse contributions from all communities, including Arab and Muslim scholars. My research has been peer-reviewed and published, and it is unfortunate to see it dismissed without thoughtful engagement. Please ensure that Wikipedia remains a platform for advancing knowledge inclusively and without discrimination. ProfMoustafa (talk) 18:50, 16 October 2024 (UTC)

I would like to clarify that the referenced article underwent rigorous peer review by two independent referees before publication. I did not pay any fees for the publication, and the journal follows a transparent policy like many other reputable open-access journals. Attempts to suppress this important discovery only serve to limit the public's access to knowledge. This work represents a significant advancement in our understanding of Mersenne primes and should be openly acknowledged as a valid and valuable contribution to the field. ProfMoustafa (talk) 07:27, 16 October 2024 (UTC)

I must clarify several key points regarding my recent contribution. Firstly, I hold a Ph.D. in number theory, and my life's work has revolved around topics like Mersenne primes. I've been an invited speaker at prestigious events, including in New York, and was honored with a full grant from the Isaac Newton Institute in Cambridge, further solidifying my expertise. Additionally, I have served as a referee for highly regarded journals, including the American Mathematical Society's Mathematical Reviews.

The insinuations in the discussion are not only misleading but also harmful. I have published peer-reviewed papers in reputable journals, including the Arab Journal of Basic and Applied Sciences. The claim that this journal operates on a pay-to-publish basis is incorrect in my case, as I have never paid for my publications there. The paper in question underwent rigorous peer review by two independent experts, confirming its validity.

I would urge Wikipedia to consider whether the person making these false claims has the relevant qualifications to pass judgment on this area of number theory. Their claims reflect a lack of understanding of the subject and undermine the credibility of Wikipedia as an open platform for verified and accurate information.

This contribution to the article is not about self-promotion but about sharing recent developments in the mathematical study of Mersenne primes, an area of significant interest. Removing such content or belittling it through unfounded accusations only serves to hinder the dissemination of knowledge to the public. ProfMoustafa (talk) 15:27, 16 October 2024 (UTC)

The inserted section has nothing to do with Lucas-Lehmer test (the SUBJECT of the article).

It is like going to the Wiki article about "Bird" and inserting a self-promoting section about "own research" in snakes and snake oil. Serge Batalov (talk) 16:51, 16 October 2024 (UTC)

=== Dear Wikipedia Community ===

It is concerning that individuals without a background in number theory, like Serge Batalov, are making uninformed claims about highly specialized topics like Mersenne primes. Given his declaration as a biologist, it is questionable how he is allowed to speak on this subject. His comments are detrimental to knowledge dissemination because he may not fully appreciate the complexity of Mersenne primes and their applications. I ask that Wikipedia take steps to prevent individuals who lack the relevant expertise from blocking accurate, peer-reviewed content.

Furthermore, I hope Wikipedia is committed to supporting diverse contributions from all communities, including Arab and Muslim scholars. My research has been peer-reviewed and published, and it is unfortunate to see it dismissed without thoughtful engagement. Please ensure that Wikipedia remains a platform for advancing knowledge inclusively and without discrimination. ProfMoustafa (talk) 18:42, 16 October 2024 (UTC)

"Given his declaration as a biologist" - and you are talking about ad nominems? That's rich.

If you don't understand the difference between biology and computational biology, then you will not understand why computational biologists just received a Nobel prize in application of AI. Serge Batalov (talk) 19:05, 16 October 2024 (UTC)

Serge,

I appreciate your recognition of computational biology, but I believe you're misunderstanding my focus. While Nobel recognition in AI applications to biology is commendable, Mersenne primes are a different field. We can all contribute to our respective disciplines without underestimating each other's expertise. It would be beneficial for everyone if you concentrated on your achievements in computational biology and AI, while those of us in mathematics work towards advancing number theory and solving conjectures like the Mersenne prime problem.

It's disappointing that you underestimated my contributions, especially given that I specialize in number theory and Mersenne primes. If computational biology is your focus, then perhaps solving Mersenne conjectures should be left to those of us dedicated to it. Blocking legitimate, peer-reviewed research undermines the core values of Wikipedia as an unbiased platform for knowledge sharing. I encourage you to focus on your field and stop blocking relevant contributions for the benefit of certain agencies or personal agendas.

Best regards,

Dr. Moustafa Ibrahim ProfMoustafa (talk) 19:17, 16 October 2024 (UTC)

Then do you understand that you are confusing scientific publishing with jamming your article down people's throats who simply came to read precisiely about Lucas-Lehmer test?

Your content is irrelevant. Stop re-posting it. Serge Batalov (talk) 20:44, 16 October 2024 (UTC)

We have already discussed your article at mersenneforum dot org. Do you recollect that? What your article does is saying: I invented the new way of calculating 10 times 10. You simply add four 25 times! How is that better? Is that new? Is that effective? No. Is it false? No. Having no practical use? Yes. Serge Batalov (talk) 20:48, 16 October 2024 (UTC)

=== Recent Developments ===

In 2024, Moustafa Ibrahim introduced a new combinatorial approach in his paper, Generalizing the Eight Levels Theorem: A Journey to Mersenne Prime Discoveries and New Polynomial Classes. The paper showed that for any prime number $p\geq 5$, and defining $n$ as $2^{p-1}$, the number $2^{p}-1$ is a Mersenne prime if and only if it divides the following expression:

$\displaystyle \sum _{k=0,k{\text{ even}}}^{\lfloor {n/2}\rfloor }(-1)^{\lfloor {k/2}\rfloor }{\frac {\prod _{\lambda =0}^{\lfloor {k/2}\rfloor -1}[n^{2}-(4\lambda )^{2}]}{k!}}$

This finding offers a fresh perspective on Mersenne primes through a combinatorial approach. — Preceding unsigned comment added by ProfMoustafa (talk • contribs) 18:52, 16 October 2024 (UTC) ProfMoustafa (talk) 05:55, 17 October 2024 (UTC)

You misleade Wikipedia and the public.

Serge Batalov deliberately misrepresents my work and exploits Wikipedia by spreading false claims. My research has been rigorously peer-reviewed by top mathematicians and published in a reputable journal in the Arab world, under the supervision of Britain. His statements are not based on the substance of the work but seem aimed at blocking knowledge, particularly targeting contributions from the Arab world. ProfMoustafa (talk) 12:24, 17 October 2024 (UTC)