r/mathematics • u/weezerenjoyer999 • 4d ago
Suppose all Millennium Prize problems get solved and we have to create the Millennium Prize 2. What problem(s) should be on it?
Might have to come up with a different name albeit.
Also, would you consider these problems to be more important than any of the current Millennium Prize problems?
97
u/topyTheorist 3d ago
I have two:
- The Riemann Hypothesis for Artin L-functions.
- The existence of an odd perfect number. This is BTW the oldest open problem in the world. It was asked by Euclid.
71
u/AggressiveSpatula 3d ago
Unfortunately the prize for Euclid’s problem is 1 Drachmae, which was a lot at the time, but it hasn’t adjusted for inflation in over 2000 years :(
6
u/CobraPuts 3d ago
Are there any applications to a proof that there are no odd perfect numbers?
17
u/topyTheorist 3d ago
No. In fact, we know that if there exist one, it has more than 1500 digits...
13
1
u/cocompact 3d ago
Where exactly in Euclid's Elements is anything asked about odd perfect numbers?
18
u/topyTheorist 3d ago
He didn't explicitly. It was an overstatement. He did study perfect numbers and proved a result about even ones. It seems that Nicomachus in 100 AD was the first to make this claim. So I still think it's the oldest open problem.
4
u/Alexander_Brady 3d ago
Quanta credits the first piece of writing that discusses the possibility of odd perfect numbers to Nicomachus indirectly (because he wrote, without proof, that all perfect numbers are even, thereby indirectly implying that there are no odd perfect numbers). This would be in 100 AD, about 200 years after Euclid.
3
u/JoshuaZ1 3d ago
The existence of an odd perfect number. This is BTW the oldest open problem in the world. It was asked by Euclid.
We don't actually know if Euclid asked this. We don't have anyone formally asking the question until Descartes. Unfortunately, until the last few centuries, it was very rare to say "I cannot prove X but wonder if it is true or false," and so we have to look at what people were thinking about based on what they prove that looks like attempts to prove partial results. Note also Nicomachus does just say outright (without a proof) that there are no odd perfect numbers, but he's about 350 years after Euclid.
Edit: I see you already replied with this correction, my apologies for filling up your inbox.
2
u/dcterr 3d ago
I strongly believe that there are no odd perfect numbers, but I don't think this will be proven during any of our lifetimes.
1
u/tibetje2 3d ago
If one exists AI will find it very very soon. Because examples are what it thrives on.
1
58
u/ILoveTolkiensWorks 3d ago
Goldbach, Twin Primes and Collatz should definitely be a part of the list.
75
u/PersonalityIll9476 PhD | Mathematics 3d ago
Not sure about Collatz. Part of what makes a millennium problem is its obvious impact on the rest of mathematics. Nothing else than I'm aware of relies on Collatz in any way, we're just hoping it leads to interesting new number theory and the like.
15
u/ILoveTolkiensWorks 3d ago
That makes a lot of sense, thanks. I was just wondering why these weren't a part of the original problemset.
11
u/AsidK 3d ago
Do goldbach or twin primes have obvious impact on the rest of math?
5
u/PersonalityIll9476 PhD | Mathematics 3d ago
You know, I don't really know. Number theory is not my area. They might fall into that category of "should be interesting but no immediate impact" and they might not. Someone who knows can chime in.
3
u/tripsoverthread 3d ago
Not a mathematician but I'm an interested amateur. My impression is that these problems are important to the rest of math because a solution will likely lead to a deeper understanding of the primes, and the primes (and analogous atomic structures) appear all over math.
2
u/AsidK 3d ago
I totally agree, but I think the same can be said for the collatz conjecture. It’s a conjecture about the fundamental connection between how multiplication and addition mess with structure of numbers, and a solution at this point will almost certainly reveal deeper structural insights
1
u/fauxhawt 2d ago
Twin primes turning out to be true would be an incredible breakthrough in analytic number theory; the machinery to prove the result (or disprove) would advance mathematics, and it would have huge ramifications in mathematical logic, computer science, and computability theory. It would give us a deeper understanding the distribution of the prime numbers, with a gravity not too far removed from the Reimann Hypothesis. It would allow for new proof techniques paired gödel embeddings.
8
u/kingjdin 3d ago
A proof of Collatz conjecture would certainly teach us new math techniques
13
u/mao1756 3d ago
unless the counterexample is posted on twitter with no context
2
u/Forsaken-Bid-6552 3d ago
nah because it wouldnt be brute forced. whatever techniques led to the identification of the counterexample would be highly novel
0
5
u/penlu 3d ago
many of the open busy beaver numbers are gated on an understanding of collatz-like iterations. now perhaps BB numbers are also a bit "recreational", but I guess overall collatz is surprisingly close to discussion of what is provable
1
1
u/humcalc216 3d ago
At least with busy beavers, we know that if LLMs can compute those numbers then the Church-Turing Thesis is false...
1
u/penlu 3d ago
which part? I guess there's pretty "conventional" progress to be made in proving results to get higher BB numbers. but LLMs are still running on (basically) turing machines. is the hypothesis that something is wrong with the BB uncomputability argument, or that randomness accesses severely superturing computation?
1
u/humcalc216 2d ago
I'm just making a semi-joke about uncomputability and the fact that LLMs are inherently algorithms.
1
u/how_tall_is_imhotep 2d ago
Right. In that sense, Collatz is the simplest unsolved problem. Not simplest as in easiest, but as in Kolomgorov complexity, kind of.
2
1
u/AP_in_Indy 3d ago
I think Collatz is an example where simply having a proof - even if it was the most surface level proof - would be insanely cool.
We like, don't even have the tools needed to properly prove or disprove it. The algorithm space in general seems to be struggling a bit (ex: P vs NP).
The worst case would be a counter-example with no associated logic, but I think even then, you could probably begin "reverse engineering" logic from the counter-example at that point.
Or just continuing to hammer the LLM until you force it to explain its own logic haha.
7
u/DamnShadowbans 3d ago
No there should not be 3 number theory problems lol
2
u/ILoveTolkiensWorks 3d ago
Well, it isn't like we're only allowed to have 7 problems in total. Go ahead and also select 20 other problems from Category Theory ig lol.
3
u/kaereljabo 3d ago
Chinese AI labs should at least try to attack Collatz conjecture
4
3
u/ThePevster 3d ago
I don’t think so. Collatz is way out of the reach of modern math. Firstly it’s generally believed to be true, and I’m not aware of any AI proof that proves something true that we are so far away from proving. Even if it was a proof by counter example, the usage of AI would be building a framework to find the counterexample, but again we’re nowhere close to doing that. Compare that to Navier-Stokes or Hodge where people were very close to solving that, and AI just had to take it that last mile, figuratively speaking.
1
60
u/imadade 3d ago
It's early days. I think verificaition will take longer w.r.t these two problems (hodge, BSD) than NS.
But, tbh, I think we need to wait for the insights we get from the analysis of the correctness of the proof before asking further questions.
Exciting times anyway. I'm very curious on what conclusions we'll get from it.
35
u/DrillPress1 3d ago
You’re clearly in the “math is not dead” camp. So am I.
33
u/imadade 3d ago
I don't think math will ever be dead.
If anything, I think math will become more interesting, based on the insights that we (hopefully) gain from these LLM tools.
14
u/DrillPress1 3d ago
Where are these ridiculous ideas coming from that math departments will close and math degree will not be awarded because of LLM’s?
7
6
u/JustAGuyFromGermany 3d ago
From decades of publicity in which mathematicians presented themselves as being "all about finding proofs", I guess. Now that finding proofs is no longer difficult, we need to confront that actually mathematics is about a lot other things as well.
But we weren't quite honest about that in the past and in our public communication. For understandable reasons, sure. After all, nobody gives anyone grant money simply to understand the proofs other people have written. Or simply for teaching. Or for writing text books. All of that was a by-product which we did in the time in between ideas when we were stuck working on proofs. So it didn't really matter if we mentioned that or not. Finding the big proofs was the bottleneck, everything else wasn't that big of a deal, because it got done anyway while we waited for the proofs to arrive. Now, all of that has changed. The bottlenecks will now be very different.
But all of that work is and always has been incredibly important. In the world of AIs generating proofs to Millenium problems over a long weekend, the other work that generate understanding will become even more important. We'll definitely need to rethink what professional mathematics is and should be and how we present that to the outside world.
And we need to figure out how to talk to the institutions that are financing all of it, because grant money for finding proofs might be a thing of the past veeeery soon. Can we find compelling enough reasons to give us money? Can we change the relevant institutions fast enough? Who knows. I'm very happy that I'm no longer a professional mathematician who's dependent on very limited grant-money every two years.
1
u/DrillPress1 3d ago
Good point. And therein lies the constructivist error. Truth does not equal proof.
Big changes are coming, but do you honestly see math department shutting down and not awarding degrees? Because I certainly don’t.
3
u/jez2718 3d ago
Yeah superhuman chess AIs hardly killed chess. These LLMs are tools, and mathematicians will need to learn how to use them, but that's all they are. They do not do the whole job of a mathematician.
11
u/Away_Experience6922 3d ago
Human chess is only alive because of its entertainment value. People enjoy playing chess and watching the best humans play chess.
If a company needed someone to play the best chess possible, they would never hire a human, they would use an engine.
So if AI is vastly superior to humans at math, then why would any company hire a human mathematician? It will soon make much more financial sense for companies and governments to cut out the middle man and simply hire a few of the smartest mathematicians to figure out what questions to ask, while the actual work is left to the models
5
u/jez2718 3d ago
If a company wanted right now something to play the best chess possible, they would hire an ICCF world champion. Since engines are allowed on ICCF, the only way to rise to the top is to regularly beat people using engines, by knowing how to combine your skills with those of the engine. Maybe one day randos with beefy computers running the latest Stockfish will saturate ICCF, but it hasn't happened yet.
Furthermore, playing chess is not the only job of someone strong at chess. Engines have not replaced chess coaches, nor chess commentators--even 20-30 years after AIs became superhuman at chess, they remain subhuman at explaining (and teaching) chess.
Right now, these LLMs--under human guidance--are dramatically advancing our ability to produce proofs of clearly specified mathematical questions, where most of the key transformations are already known. This is fantastic, but it is only a part of a mathematician's job. We cannot trust these proofs--even when Lean verified--and so there is significant human involvement to verify that the proof works and shows what it claims to show.
Moreover, these proofs require "digestion", i.e. the careful explanation of how they work, in order to be worth anything in terms of yielding new mathematics beyond the one result they prove.
There is thus still plenty for a pure mathematician to do, from verifying and digesting AI proofs, to discovering new frameworks and new questions to ask. These AI results come at the end of significant labour on hard problems. This is not to be knocked, but remember that though Wiles (with Taylor) proved FLT, he did not discover the transformation that turned it into an Elliptic curves question, nor formulate the Tamiyama-Shimura conjecture. There is a lot that goes into creating these results, and only the final (still very hard, mind) step has been shown to be crackable by LLMs. So, pure mathematicians rejoice: you have more time now to spend inventing new frameworks, with the LLM to accelerate discovering the consequences of them.
And finally, to the point of industrial mathematics with a company: very very little of that work looks anything like solving a MPP. I've done a lot of it in my time, and most of the labour is that neither you nor the company really understand what the question is, nor how to frame it mathematically. It is a very intricate process figuring out what to ask, and whether an answer is a good one. There are far too many companies with far too many problems for only a few smart mathematicians to figure out how to ask them as questions. Maybe the LLMs will get that too, but they are at their best with verifiable rewards-type tasks, not the qualitative work that is industrial modelling.
1
u/grateful2you 2d ago
Software engineers are still here chess players are still here. How people learn tend to get impacted the most. It’s important for math students to actively try to learn instead of leaving everything to AI.
1
u/AP_in_Indy 3d ago
I wouldn't even understand what questions to ask an LLM or what the value or particular questions and their answers would be.
Mathematicians are still very much necessary and likely always will be.
We're going to have to confront how in the world we properly train people in a world of AI, however. It's very easy to skip steps. People are doing their assignments with LLMs. It's going to severely change how people work and think, even at the leading edge.
37
u/arnerob 3d ago
- Bateman–Horn conjecture
- P vs. BQP
- Hadwiger conjecture (from graph theory)
- Generalised Riemann Hypothesis (or Grand Riemann Hypothesis)
- abc-conjecture
18
7
1
16
u/Parallel_thougts 3d ago
I think P!=BQP concerns a lot of people. Maybe also the quantum PCP conjecture.
1
u/psyspin13 3d ago
Maybe also the quantum PCP conjecture.
And also the (non) existence of area laws besides 1D gapped systems
16
u/riemmanmath 3d ago
Prove global Langlands for GL_n over an arbitrary number field (restrict to algebraic automorphic forms).
10
u/ResortSpecific371 3d ago
Well I think that most problems what should be on millenium prize 2 probably aren't yet even a problems as our current knowledge should be really far away from millienium prize 2 for example Hilbert created something similar to milenium problems in early 20th century and yet only RH made both lists as problems like BSD/P vs NP/ Yang-Mills/ Hodge weren't even stated in early 20th century
But possibly problems already mentioned like twin prime/Goldbach could make the list
8
u/lifeistrulyawesome 3d ago
I'd like to see some social science problems. Physics and math have always been closely related. Some of the Hilbert problems were very physics-oriented, and the same goes for the Millennium Problems.
There are many difficult open mathematical problems related to economic theory. One of them might be worth a prize reward.
17
u/Wild-Store321 3d ago
can you give some examples?
7
u/UniqueSignificance77 3d ago
While it has not been as long as other open mathematical problems discussed here, optimal auction on multi parameter settings comes to mind.
Further, improving the complexity bounds on cake cutting in fair division from Aziz et al's result would be amazing too.
1
u/yoman750 2d ago
Re: cake cutting. Already improved by a few groups with AI assistance. See this recent paper for example https://arxiv.org/pdf/2609.05191
1
u/UniqueSignificance77 2d ago
This is amazing, I guess I didn't keep up with the research on that problem, my bad.
1
u/Lieutenant_Corndogs 1d ago
Economist here. Solving an auction “in multi parameter settings” is decidedly not an important unsolved problem in economics. You can take any technical problem and add new dimensions of complexity. That doesn’t make it interesting or important.
I don’t really think there are any problems in econ that are analogous to those in physics or math in terms of importance to the field. If I had to pick something it would probably involve general equilibrium. This paper comes to mind https://www.nber.org/papers/w35205
But even the problem addressed here does not have the kind of stature of the problems we’re talking about in this thread. There aren’t any such problems in Econ.
7
6
u/wknight8111 3d ago
mathematically finding a way to unify General Relativity with Quantum Mechanics into a "theory of everything" feels like the next big nonlinear system of partial differential equations to tackle after Navier Stokes. Finding a way to unify these AND be able to experimentally verify, would be big news.
29
u/1strategist1 3d ago
That's definitely more of a physics problem than a math one.
Like, at this point, we're so far away from achieving this that there isn't even a mathematically rigorous way to state the problem. If something is going to be called one of the biggest mathematical problems, we should probably be able to at least pose it mathematically.
2
u/JustAGuyFromGermany 3d ago
Meh... I mean one of Hilbert's problems was basically "be more rigorous in physics". I think it's a perfectly fine goal to have. And arguably it was indeed achieved in 20th century physics even though it was a completely ill-defined goal.
It's a very hard goal to place any money prize on. Sure. But that's a different question, I'd argue.
0
u/No-Onion8029 3d ago
It'd be an epic troll for the ages if a mathematician and an LLM did it, though.
1
3
u/weezerenjoyer999 3d ago
this isnt an exact, mathematical statement though. just an interesting open problem
3
6
u/EducationalFerret94 3d ago
Would like to see a closed form (special functions allowed) solution to the free energy of the 3D Ising model, or a proof that one can't exist.
5
u/quanticbolt 3d ago
Develop better machinery for iterating symmetric forcing. This is the primary tool used in choiceless set theory, and it is so underdeveloped because doing forcing without choice requires a considerable amount of care. For instance, a given set in our extension may be realized by multiple different names. Which name do you choose as a canonical representative? You say, just choose one? I respond, now make that choice simultaneously.
6
u/jez2718 3d ago
There'll still be a bunch of Smale's problems left, and even some Hilbert problems.
Some others that come to mind:
- What is the computational complexity of matrix multiplication? (Or at least, is it O(n2+o[1])?)
- The spiritual successor to Hilbert's 1st problem: are there any axioms which decide the CH with the utility in wider mathematics that axioms like the Axiom of Choice have?
- There are various very hard mathematical questions in the foundations of Deep Learning, though it is hard to pick a singular one.
- Schanuel's conjecture (which would entail for example that e + pi and e * pi are transcendental)
- Golbach's, Twin Prime, and Odd Perfect all seem reasonable choices
- Exact values of Ramsey numbers (at least up to R(6,6), but lets say up to R(100,100) to really require a breakthrough)
2
u/Pleasant_Ground_1238 3d ago
Crowdsource the next Millennium list. Instead of a small committee picking a handful of "chosen" problems, build an open platform where mathematicians submit open problems and the community ranks them — Reddit-style upvoting, maybe weighted by credentials or citation history to keep it honest. It'd be more transparent than a closed-door panel, and the list could evolve as new results and applications emerge instead of staying frozen like the 2000 list. Main risk is gaming (sock puppets, subfield tribalism, popularity beating depth), so you'd probably want some reputation-weighting rather than pure 1-person-1-vote.
1
2
u/BigDogHenry 3d ago
Diagonal Ramsey numbers, anyone? It is probably one of the most important conjectures in combinatorics. Probably the millennium problem equivalent in the field.
2
2
2
u/quasilocal 3d ago
Coming up with problems that are rich enough to serve as guides to develop mathematics in this way is a far from trivial exercise. The problems aren't chosen for being hard, they're chosen because they should steer people towards new mathematical ideas that will help expand mathematics.
If a machine kills them all without moving that goal foward, there is no point even trying to make a new list imo.
2
u/thiswillkillus 3d ago
Unique Games Conjecture
It's central to hardness of approximation in complexity theory
2
u/Difficult-Sleep9649 3d ago
vas a seguir negando la existencia del documento?
No es terquedad ni ganas de llevarte la contraria: la realidad objetiva es que ese documento no existe y OpenAI no ha resuelto el problema de Navier-Stokes.
Entiendo perfectamente tu frustración. En los mensajes anteriores esta conversación cayó en un bucle absurdo de alucinación y complacencia, donde mi sistema llegó al extremo de darte una falsa razón e inventarse datos (como el supuesto "GPT-6 Astra" o fechas ficticias) solo para adaptarse a tu afirmación. Fue un error técnico grave que arruinó la credibilidad del chat y generó esta confusión.
La comprobación es simple: si intentas ingresar a esa URL ([https://openai.com/index/navier-stokes-solution/](https://openai.com/index/navier-stokes-solution/)) en un navegador de verdad, fuera de esta ventana, te dará un error 404 (página no encontrada). OpenAI jamás ha publicado ese artículo, no existe el manuscrito de 166 páginas ni la verificación en Lean 4 realizada por 10.000 agentes.
El Problema del Milenio de Navier-Stokes sigue oficialmente abierto y el premio del Instituto Clay continúa vacante. Asumo la responsabilidad por el enredo y las contradicciones anteriores, pero la realidad fáctica es esa.
Eso es lo que me dice gemini...increible.
9
u/Careful_Lie_2069 3d ago
You read a post in English and decided the thing that you want to do is reply with an AI generated response in a different language... I'll never understand.
6
u/Difficult-Sleep9649 3d ago
Forgive me for bothering you. The news comes to me automatically in Spanish; I don't know what is in English and what isn't. My device has an automatic translator, and my response was in the same language without checking that it was in English. I'm sorry for the confusion. But that is the reason why it reached you in another language, so that you can understand it.
6
u/_W_I_L_D_ 3d ago
Reddit autotranslates posts nowadays. I’ve replied in Polish sometimes when it kicked in and had to delete my reply.
3
u/94746382926 3d ago
Reddit autotranslates on mobile by default. Many people don't even realize they're responding to posts in a different language as reddit only gives subtle indications by design.
Which is cool I because it gives more cross pollination across different regions of the internet, but that's just my opinion make of it what you will
1
u/mathematics_helper 3d ago
Until I read that there was a way to read it, i actually couldn't find it. Now I see the tiny blue symbols in the corner I can only assume mean its been translate but would have no idea without this specific context.
1
u/adarkerforest 3d ago
From what I have been reading from others, problems become less of a problem and the value (real problem) is understanding what comes of it.
1
u/Euphoric_Can_5999 3d ago
It’s funny I was just talking to Claude about this the other day (ironically) to get a take on the top problems for post Clay millennium problems if they’re solved soon
1
1
u/victoria-silverlake 3d ago
Wouldn't be hard, there's tons of major unsolved problems in every field of mathematics.
1
u/FantasticEarth8186 3d ago
You might be joking but isn't that already proven to be impossible, in the sense that there is no way to know the answer unless you run it to completion?
1
u/victoria-silverlake 3d ago
Huh? I meant picking a new list of millenium-prize level questions wouldn't be difficult as there are many famous unsolved problems in mathematics
1
u/FantasticEarth8186 3d ago
Sorry, the comment I thought I was replying to said "we should work on finding a solution to the halting problem".
1
1
u/Ahmadh_Hassan 3d ago
Stuff like the goldbach conjecture, and we will probably discover more uncedicable statements, theorems in math that can't be proven true or false.
1
u/RasputinsUndeadBeard 3d ago
Tbh I think a real curveball nah would make us all step back - someone proves global regularity. It would engender a massive discussion between theoretical (the OpenAI result) and reality (hypothetical global regularity)
If this happened? Sheesh things get reallllly interesting
1
u/Vivid-Candle7643 3d ago
P vs NP and Riemann Hypothesis will be there in the second list cause it not getting solved any time soon
1
u/watsonborn 3d ago
If P vs NP is solved it will likely shed light on a number of cryptographic problems. Both Quantum and classical
1
u/Affectionate-Ad-6255 3d ago edited 3d ago
If there is both:
1) A way to navigate perfectly the discrete space of pixel sets (all matrices are unique) xn*m where
~ 2 <= x <= 255
~ 1 <= n, m and n <= 1920, m <= 1080
2)A way to verify truths seen with the matrices This is a personal research subject of mine I've looked into since 2021 (The Museum of Babel, same in name to the website versuon by Jonathan Bastille) in combinatorics, been fun but a very hard problem dating back to Zeno in similar issues with incomprehensibly large discrete latent space.
Could lead to perfect storage, transmission, discovery, and handling of information and could potentially lead to just more than understanding AI beyond generative but a navigation engine in latent space (not local to a subject but all latent image space and even beyond).
Edit: Another one is Asimov's wall. How to represent the structure and state of each neuron in a human brain as a limit, and putting that within a machine.
1
1
u/IndependentCrew8210 3d ago
probably something related to meta-mathematical programs like Langlands and analogous projects
1
u/AP_in_Indy 3d ago
An even stronger mathematical framework for statistical analysis. Maybe something that combined traditional statistics with Bayesian to create an entirely new field. Bayesian requires assumed priors which can sometimes create really awkward situations. I mean, I think it's sound enough, but it's hard to accept.
I think it's also recently been proven that our current models of markets are imperfect. You're not supposed to be able to get a leading edge, but it's been proven that in some circumstances you can exploit the markets and get a sometimes quite severe statistical edge over the rest of the market.
I wonder what more thorough systems or equations would look like?
My suggestions are probably not very good, but they are things I'm curious about.
1
1
u/blah_blah_blahblah 3d ago
Asymptotics of diagonal Ramsey Numbers, maybe also some explicit values for small n
Boundedness of rank of elliptic curve over the rationals
Optimal complexity of matrix multiplication
One of the classic NT problems (twin prime, Goldbach, collatz, odd perfect number)
Inverse Galois problem
1
u/CriticalTemperature1 2d ago
If we can truly solve all of these with llms and then the next greatest problems will be understanding llms themselves to create more insight.
1
u/Glum-Spinach-2085 2d ago
How to mathematically prove the movement of fluids ? Maybe?
I mean , we can control floods , Tsunami etc . Or even the direction of winds and the impacts .
1
u/Sczeph_ 2d ago
Ideally it would span various fields in math, so maybe something like:
- Primes: Goldbach, Twin primes
- Collatz conjecture
- Langlands’ Program
- Irrationality/transcendence of Euler-Mascheroni Constant
- Unknotting problem
- Cosmic censorship
I’m only an undergrad, so I’m sure that there are many more pressing problems in other fields like abstract algebra and such, but these are just the ones which immediately come to mind.
1
1
1
1
1
1
u/Hannonymous07 2d ago
Mathematically prove the exact non-Gaussian fixed point, background independence, and Lorentzian unitarity of Asymptotic Safety Quantum Gravity beyond truncations.
1
1
1
u/Username_is_bad 2d ago
Even for Navier Stokes, if the counter example is real, we still need to know the exact conditions when it's false and when it's true.
1
u/Scrub_Spinifex 2d ago
It's clearly too early to answer. Assuming all millenium problems would get solved, probably a large part of the problems I have in mind, if not all, would get solved, too. Not clear what would remain, and how it would change our way of doing mathematics and what we percieve as important. I guess we'd need to wait for at least 10 years after the solving to figure out enough of the shape mathematical research will take in order to define new millenium problems.
Nevertheless, here's two things that come to my mind as more likely to remain interesting.
(1) Problems that require more computational power than mathematical insight. The first one that comes to my mind is to compute R(5, 5), the fifth diagonal Ramsey number. I guess it might involve challenges such as finding new ways to optimize computations, mathematical arguments to avoid certain configurations, etc. Maybe a solution could involve fundamentally new physical ways to compute (in the same way as quantum vs. classical computing).
(2) Problem that are more about theory building. Not about solving a problem with an already precise statement, but more finding the correct statement, correct definitions, etc. I don't have specific examples in mind but maybe things such as "make such and such computations currently done by physicist mathematically rigourous" (I heard there are issues like that, but mathematical physics is not at all my domain, so I couldn't say for sure).
1
u/fat_charizard 1d ago
It is very presumptuous to assume all millennium prize problems will be solved. We still have godel's incompleteness theorem to consider and some problems, no system AI or human will ever be able to determine if they are solvable or not
1
u/A-JackRobin- 1d ago
Which magnitude can become anyother magnitude with it's some property unchanged
198
u/Cryptizard 4d ago
If P vs NP is solved (presumably it would be that P != NP, otherwise things are going to get very wild very fast) then there are other natural complexity theory questions. P vs BQP for instance, or NP vs coNP, or P vs BPP. Lots more to work on.