Talk:Millennium Prize Problems

Undecidable?
Is the prize only awarded for showing that the conjectures are true or false, or does one also claim the prize for showing that a conjecture cannot be proven via the axioms? —Preceding unsigned comment added by 86.153.11.178 (talk) 19:50, 24 January 2011 (UTC)
 * The Clay Mathematics Institute's official policy gives them leeway in determining what constitutes a solution. As mathematicians generally accept "this is independent of ZFC" as a solution (see Hilbert's problems) it seems highly likely that they would reward such a proof.
 * This question puts me in mind of another however: should the potential insolubility of these problems be mentioned? For example, if P = NP then there is a proof of that fact. However there is no particular reason to believe that the Birch and Swinnerton-Dyer conjecture must have a proof. — Preceding unsigned comment added by Stellaathena (talk • contribs) 16:15, 3 December 2020 (UTC)

It is all prone to attack and very well all problems in the set are solved in a single swipe
http://meami.org/

This is the html version of the file http://claymath.org/library/monographs/MPPc.pdf. Google automatically generates html versions of documents as we crawl the web.
 * <...WP:COPYVIO material removed by User:EdJohnston. See below...>

—Preceding unsigned comment added by 204.128.192.4 (talk) 05:50, 12 May 2010 (UTC)
 * The material originally posted by the IP editor above comes from a PDF file that is still available online from the web site of the Clay Mathematics Institute. http://www.claymath.org/library/monographs/MPP.pdf, whose home page is at http://www.claymath.org. Verbatim reproduction of this file on Wikipedia surely violates their copyright, so I've deleted it from this talk page. Our readers should consult the original PDF file instead. EdJohnston (talk) 19:28, 29 June 2010 (UTC)

Millennium Prize Problems Conditions
"Only the Poincaré conjecture has been solved, but the solver has not pursued the conditions necessary to claim the prize." What exactly are these conditions?

EDIT: [the french article says] the solution must be widely accepted by the mathematical community for two years after publication before the Millennium Prize is given away.


 * The conditions to be awarded the prize are generally given as three stages. Above is 'stage 2'.
 * 1-"Before consideration, a proposed solution must be published in a refereed mathematics publication of worldwide repute..."
 * 2-"..and it must also have general acceptance in the mathematics community two years after."
 * 3-The science advisory board of the Clay Mathematics Institute appoints a panel to inspect the problem and make a report and recommendation the the directors of the Institute who in turn have the final say in awarding the prize.
 * {info from the book Poincares Prize by George G Szpiro} Zybthranger (talk) 00:20, 7 October 2008 (UTC)


 * thanks, but the perelman article states that "he has yet to be offered the prize," meaning that it seems only to be a matter of time. Twipley (talk) 16:05, 9 February 2009 (UTC)

Perelman was offered the prize but declined it. A personal choice that all should/must respect, as it doesn't affect anyone else if he accepts it or not.

There should be more than money to incentivize the mathematician. But I've yet to determine what it is....

Many famous problems have already been solved, but why should the solver tell anyone? — Preceding unsigned comment added by 66.223.207.98 (talk) 23:47, 22 November 2018 (UTC)

Because it's the right thing to do

Navier Stokes
I was told during my time at grad school that the rules were changed so that giving a counter example existence and uniqueness for Navier Stokes problem specifically was not sufficient to win the prize. Is this officially the case? Thenub314 (talk) 16:10, 21 January 2009 (UTC)

In the case of the P versus NP problem and the Navier-Stokes problem, the SAB will consider the award of the Millennium Prize for deciding the question in either direction. In the case of the other problems if a counterexample is proposed, the SAB will consider this counterexample after publication and the same two-year waiting period as for a proposed solution will apply. If, in the opinion of the SAB, the counterexample effectively resolves the problem then the SAB may recommend the award of the Prize. If the counterexample shows that the original problem survives after reformulation or elimination of some special case, then the SAB may recommend that a small prize be awarded to the author. The money for this prize will not be taken from the Millennium Prize Problem fund, but from other CMI funds. Never mind their rules are clear enough about counter examples. Thenub314 (talk) 16:16, 21 January 2009 (UTC)

Copyright issues
The entire lead of this article is found verbatim at http://www.claymath.org/millennium/ and there may be more within the content. The article needs some review as to these matters. My76Strat (talk) 00:04, 1 June 2010 (UTC)

Copyright problem removed
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P vs NP
I have removed the references to Deolalikar's briefly-claimed proof, since, following many flaws found in the community, he has removed all mention of the draft paper from his web page. I know that at least one of the other MPs had a briefly claimed proof, later withdrawn, so clearly there is no intention to record all such here. Should a corrected version appear, then of course it should be mentioned here, but currently that seems vanishingly unlikely.Educres (talk) 21:57, 10 August 2010 (UTC)

What if Riemann Hypothesis is Baseless?
Riemann Hypothesis is baseless, where it can neither be true nor false. http://www.gsjournal.net/Science-Journals/Essays/View/4491 — Preceding unsigned comment added by 112.134.176.24 (talk) 19:35, 30 January 2013 (UTC)

Declined in 2010?
"but he declined the award in 2010" - That's unsourced, and I know for a fact it's incorrect - I read about him declining the award while I was still in college, and I graduated in 2009. It must have happened around 2008 or 2009, definitely not 2010. BlueRaja (talk) 00:06, 11 March 2013 (UTC)


 * The prize was not even offered until 2010. See . Roger (talk) 04:21, 12 March 2013 (UTC)


 * He declined the Fields medal in 2006. You may have been thinking of that. 50.0.121.102 (talk) 21:16, 23 January 2014 (UTC)

P = NP Solved by Physics Trick
What do you think of the use of a fact from physics to sove a mathematics problem? As v approaches c, NP computational scaling factors approach polynomial ones, for example, if x is the amount of data to be calculated and it can require either x^2 time (or rate) or x! time, then using Relativity in the form of the Twin Paradox:

x! / x^2 = (1- v^2 / c^2)^0.5

Solving for v:

v = c (1 - (x^2 / x!)^2

Albert deBroglie (talk) 23:55, 1 September 2013 (UTC)
 * This is discussed a little in section 6 of Scott Aaronson's article "NP-complete problems and physical reality". Aaronson and Watrous have another paper saying that in a computational model allowing computing in closed timelike curves (from general relativity), PSPACE becomes tractable.  The formal P vs NP problem just counts the number of steps performed in the calculation though: physical time distortions don't change that. 50.0.121.102 (talk) 21:22, 23 January 2014 (UTC)

Navier-Stokes
There's a new paper by Otelbaev purporting to solve NS, that's currently mentioned in this article both in the NS section and in the lede. The paper seems to be getting a fairly subdued reception in the math world, so I'm about to take it out of the lede but leave it in the NS section. If anyone really feels differently, feel free to put it back. 50.0.121.102 (talk) 21:55, 23 January 2014 (UTC).

It seems like this one is on its way to solution - https://www.jpost.com/HEALTH-SCIENCE/The-Israeli-who-solved-a-200-year-old-math-equation-568729

Removed confusing text
In the section "Yang–Mills existence and mass gap", the article used to say:
 * The official statement of the problem was given by Arthur Jaffe and Edward Witten & recent status by Michael R. Douglas.

The latter part of this sentence looked suspicious, given the ampersand, and didn't make sense to me. Despite the fact that the "information" has been in the article for almost 2 years, given that I have removed that part of the sentence. If someone can clarify the connection to Douglas, and it seems relevant to mention in the article, please do so. - dcljr (talk) 22:04, 2 July 2014 (UTC)
 * 1) neither "Yang–Mills" nor "mass gap" appear in the Michael R. Douglas article (although it does connect him to the Clay Mathematics Institute),
 * 2) "Douglas" does not appear anywhere in the Yang–Mills existence and mass gap article, and
 * 3) the editor who  to the article has been blocked indefinitely for disruptive editing,

"stated" versus "chosen"
In the opening sentence I suggest replacing the word "stated" by "chosen". The Clay Institute did not state these problems. They chose them from well known mathematical problems that were all stated earlier by others, in some cases more than a century earlier. — Preceding unsigned comment added by Mosher (talk • contribs) 14:45, 17 September 2015 (UTC)

Aaronson Quote Absurd
In the P v. NP section, the Aaronson quote:

"Everyone who could appreciate a symphony would be Mozart..."

...is an opaque and absurd exaggeration on the consequences of P = NP. Could we get a quote that isn't trying to be poetic, or strike the quote entirely? TricksterWolf (talk) 21:43, 10 December 2015 (UTC)

External links modified
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Hi, you can find some interesting facts in the preprint https://www.researchgate.net/publication/343509578_Three_Ill-posed_Millennium_Prize_Problems_with_an_Empty_Nonunique_or_Incomplete_Sets_of_Solutions?_sg=CFD90EtHRTQv_Kd-by-8C-tEIGXs3z7NKEwNZp318BLIR_CWn1QFzn6CY2FxdfDwS847jJseVGVbKXw — Preceding unsigned comment added by 70.29.210.81 (talk) 06:05, 19 January 2022 (UTC)
 * Added preprint

Yang-Mills equations
The section says: "As a classical field theory it has solutions which travel at the speed of light ..." This is factually incorrect, the correct statement is "At high energy, or as the coupling g -> 0 the equations are reduced to Maxwell's and the propagating quanta look like free particles similar to photons, and are called gluons." I will watch this space for commentary. Kotika98 (talk) 04:28, 14 June 2017 (UTC)


 * That is correct and should be fixed. As it has been shown by lattice computations, the theory at low-energies develops a mass gap for the force carriers making them massive.--Pra1998 (talk) 06:11, 14 June 2017 (UTC)

Poincaré conjecture proof first published in arxiv.org?
From what I know, Grigori Perelman first published his proof(s) on arxiv.org. If this is true, I think it would be important to mention this on his page (because I think this highlights the importance of pre-prints). So far, this is what I have found. Any help to resolve this would be appreciated. https://www.claymath.org/millennium-problems-poincar%C3%A9-conjecture/perelmans-solution But I don't see a link to the proof directly here. Searching for the author I get this arxiv.org user which I think is the author. https://arxiv.org/search/math?searchtype=author&query=Perelman%2C+G This Youtube video (I know this is not a good source) claims that it was in fact first published in arxiv.org which is why it was picked up so quickly. https://www.youtube.com/watch?v=GItmC9lxeco Itchyjunk (talk) 22:57, 23 May 2019 (UTC)

Claims of a proof - what to do until matter is finally concluded?
When do claims of a proof merit a mention?

Don't want to mention some random dude claiming he has a proof. So probably want something peer reviewed before mentioning it?

Eg. timesofindia: claims Riemann proved indicates a committee considered 7 reviewers. Should this be regarded as adequate peer review? Does this sort of thing merit a mention? Should we be wary of things like this perhaps especially when the reports indicate "there was a reluctance on the part of the editors of international journals to put the paper through a detailed peer review". C-randles (talk) 13:58, 30 June 2021 (UTC)


 * The right approach is to avoid any citation until the community at large will accept the proof, exactly as happened for Perelman. Just remember that it is the Clay Institute to decide in the end. Until then, this is just a claim as many others recurred in the course of time.--Pra1998 (talk) 14:34, 30 June 2021 (UTC)

Rieman Hypothosis Diagram
The diagram attached to the Rieman Hypothosis is either misleading or disproves it. As I doubt it disproves it, it should be removed. Dstar3k (talk) 02:36, 17 June 2023 (UTC)

2 dimensional sphere?
in the solved problem it talks about a 2 dimensional sphere. Is this a mistake and it is supposed to be a circle, or am i missing something? Maybe it should be rewritten to be more clear. Chuck541 (talk) 21:06, 15 July 2024 (UTC)


 * It could be clarified, but there’s no mistake. In mathematicians’ parlance, the two-dimensional sphere is the usual sphere you think about as living in three dimensions. The idea is that the sphere, in and of itself, has only two degrees of freedom. You just see it inside three dimensional space. Gumshoe2 (talk) 00:02, 16 July 2024 (UTC)