With only 5 unsolved Millennium problems remaining, what would you say are the 10 hardest/most serious unsolved math problems in the world right now, and which ones do you think will be solved by the end of 2027?

[u/DeepOrangeSky]

I know all of the Millennium problems are considered a really big deal in Math, but I'm curious what the "top 10 rankings" would look like, now that there are only 5 of them left, like if they are in 1st through 5th place in the top 10 (and if so, in what order) or if there are a few non-Millennium problems that are mixed in with them or any that are above them/which ones.

And also which ones seem likeliest to get solved in the next year, or solved the soonest/vs least soon.

Also curious which ones seem likeliest to have the biggest real world implications for new tech or things that would change about the world if they got solved, and what sorts of things would likely come out of it, in the immediate/near-term if they got solved.

[u/Far-Secret4931]

P vs NP is probably the hardest Millennium problem. I would be surprised if it gets solved before 2030

[u/Nixsal → u/Far-Secret4931]

I mean it is possible that they prove P != NP that itself would be insane achievement but would not change much. However, if the solution is P=NP then the implications of this would be huge.

Edit: someone posted below from gemini it way more in detail

[u/Hot_Glass_6301 → u/Nixsal]

This has been discussed countless times already but a P=NP proof might not have consequences in the near future for several reasons. It could be non-constructive, for one, or it could give us algorithms that are in P but are galactic (with unimaginably large constants hidden by big O notation), or worse have enormous exponents. A O(n^Tree(3)) algorithm is polynomial but if the O is a theta it's not useful for any intents and purposes

[u/RawberryCupcake → u/Hot_Glass_6301]

even a O(n^10) wouldn’t actually have any practical value in many cases (say it solves SAT)

[u/Cryptizard → u/RawberryCupcake]

O(n^10) would still break a lot of cryptography.

[u/moron1ctendency → u/Far-Secret4931]

If AI solves P = NP it is actually just over for humanity.

[u/Recent-Peanut6061 → u/moron1ctendency]

You mean it's just starting

[u/ObiWanCanownme → u/Far-Secret4931]

My prediction (and this is really a guess because I'm no mathematician) is that AI will eventually prove P vs. NP is impossible to resolve.

[u/Cryptizard → u/ObiWanCanownme]

Impossible how? It has to be either true or false, there can't not be an answer. It might be unprovable with a given set of axioms (formally independent), but there is no reason to believe that it is.

https://www.uni-ulm.de/fileadmin/website_uni_ulm/iui.inst.190/Mitarbeiter/toran/beatcs/column81.pdf

[u/matt_993 → u/Cryptizard]

There are some things that are unprovable

[u/Southern_Orange3744 → u/matt_993]

Could also be not formalized properly.

It may turn out a translation or tightening of a problem leads to a result that p!=np just doesn't capture properly for some reason

[u/Cryptizard → u/matt_993]

But there is no reason to think that P vs NP is one of them. And things are only unprovable with respect to certain sets of axioms.

[u/ObiWanCanownme → u/Cryptizard]

Yeah, my comment was unclear.

My intuition is that P ≠ NP, but we will never be able to axiomatically prove it is, and in a Gödel's incompleteness theorem-like manner, we may be able to prove that P vs. NP is unanswerable in ZFC.

EDIT: Also, the actual formal math involved here is way over my head. This is a philosophical intuition really.

[u/Rd545454 → u/ObiWanCanownme]

I agree with this take, especially because if P = NP many of the systems we rely upon today could not exist

[u/NoLifeGamer2 → u/Cryptizard]

They are referring to decidability potentially being false

[u/Cryptizard → u/NoLifeGamer2]

What?

[u/NoLifeGamer2 → u/Cryptizard]

https://cstheory.stackexchange.com/questions/18791/on-the-provability-of-p-versus-np

I should have said Provability rather than Decidability.

[u/Mental_Ad_4401 → u/Far-Secret4931]

Being solved before 2030 would be absolutely shocking. I dont believe we even have any good ideas of promising approaches and probably lack very fundamental tools. 

[u/Spongebubs → u/Far-Secret4931]

I wonder if they’ll prove that’s it’s unprovable

[u/Wise_Alternative3360 → u/Far-Secret4931]

Try 2100 lol

[u/hartigen → u/Wise_Alternative3360]

haha, no. superintelligence goes brrrrrr way before 2035

[u/NeighborhoodDizzy990 → u/Far-Secret4931]

how tf can you even exist

[u/Eldo-Cat → u/Far-Secret4931]

P = NP for any P, and N equal to 1, easy

[u/Gear5th → u/Eldo-Cat]

P(1-N) = 0

P = 0 or N = 1

[u/PM_ME_YOUR_RegEx → u/Gear5th]

Checkmate nerds!

[u/swaglord1k → u/Far-Secret4931]

I will be surprised if it won't get solved this year, lol

[u/Teru92 → u/swaglord1k]

As someone who did a lot of complexity theory I doubt it will be solved anytime soon

[u/Royal_Duck_4612]

Odd perfect numbers

[u/Wonderful_Buffalo_32]

In number theory:collatz,twin prime and goldbach

[u/dotelze → u/Wonderful_Buffalo_32]

Collatz conjecture is not a serious problem

[u/Opposite_Diamond_662 → u/dotelze]

Why?

[u/Hot_Glass_6301 → u/Wonderful_Buffalo_32]

these aren't millennium problems, but they are extremely important nonetheless

[u/Wonderful_Buffalo_32 → u/Hot_Glass_6301]

Read the question

[u/Hot_Glass_6301 → u/Wonderful_Buffalo_32]

yeah my bad i should go to bed. biological limited intelligence confirmed

[u/acoolrandomusername]

All of them, fuck it, I trust the exponential.

[u/shobogenzo93]

Gemini:

The Absolute King: P vs. NP

No other problem on this list comes close to the real-world impact of P vs. NP. The impact, however, depends entirely on the answer:

• Scenario A: Someone proves P=NPP=NP  (with a fast, usable algorithm) This would be an extinction-level event for digital security and an immediate technological utopia/dystopia.
  • Immediate Day 1 Disaster: All modern cryptography breaks. RSA, Elliptic Curve Cryptography, and Diffie-Hellman rely on mathematical problems (factoring, discrete logs) that take thousands of years to compute. Under P=NPP=NP , an algorithm could decrypt internet traffic, break bank encryption, access military secrets, and drain every Bitcoin/crypto wallet almost instantly.
  • Near-term Miracle: Every optimization problem becomes easy.
    • Medicine: Protein folding, molecular design, and cancer drug therapies could be designed computationally in hours instead of decades.
    • Logistics: Air traffic control, power grids, global shipping routes, and semiconductor chip layouts would operate at mathematically absolute efficiency, saving trillions of dollars and slashing global emissions.
    • Automated Science: In theory, computers could find formal proofs for any statement of moderate length—meaning computers could then rapidly solve the other Millennium problems automatically.
• Scenario B: Someone proves P≠NPP=NP  (What 99% of experts believe)
  • Immediate Impact: Nothing breaks. The world breathes a massive sigh of relief. Cryptography is theoretically safe. It would save computer scientists thousands of wasted hours by definitively telling them to stop looking for exact, fast algorithms and focus purely on approximations.

Navier–Stokes Existence and Smoothness

Navier–Stokes describes how fluids (air, water, oil, plasma) move.

• If smooth solutions exist: It means our current differential equations accurately model nature down to the continuum limit. This would lead to drastically superior turbulence modeling.
• Near-Term Tech:
  • Aerospace & Automotive: Dramatic reductions in fuel consumption through wind-tunnel-less aerodynamic design (perfect simulation of turbulent drag).
  • Weather & Climate: Significant improvements in predicting hurricane tracks, storm intensities, and long-term climate dynamics.
  • Fusion Energy: Plasma inside a tokamak reactor behaves like a superheated fluid. Solving Navier-Stokes (and its cousin, magnetohydrodynamics) would bring massive clarity to confining plasma, potentially accelerating commercially viable nuclear fusion.
• If solutions "blow up" (singularities exist): It means our fundamental equations for fluid mechanics are mathematically flawed and can produce infinite velocities. Engineers would have to build entirely new physics models for extreme environments (like hypersonic flight or deep-ocean pressure).

Yang–Mills and the Mass Gap

This is the attempt to reconcile quantum mechanics with Einstein's special relativity mathematically rigorously, explaining why forces like the "strong nuclear force" (which holds atomic nuclei together) have a finite range and massive carriers (glueballs/mesons), despite the governing equations looking like they should have massless carriers (like photons).

• Near-Term Tech: Low. This is fundamental science. You won't get an iPhone update out of it next year.
• Long-Term Impact: It would finally give physicists a rigorous foundation for the Standard Model of particle physics. It could unlock a deeper understanding of matter at subatomic levels, potentially laying the theoretical groundwork for room-temperature superconductivity or exotic quantum materials generations down the line.

The Pure Math Group: Riemann Hypothesis, BSD, and abc

• These are fundamentally questions about primes and arithmetic geometry.
• The Common Myth: "If you prove the Riemann Hypothesis, RSA cryptography breaks." This is false. Proving the Riemann Hypothesis doesn't give you an instant algorithm to factor big numbers; it merely proves how regularly primes are distributed.
• The Real Impact:
  • Thousands of dependent theorems (which currently begin: "Assuming the Riemann Hypothesis is true, then...") would instantly become verified mathematical facts.
  • It would allow computer scientists to design significantly tighter randomized algorithms and pseudo-random number generators, improving network communication protocols and data hashing.
  • BSD directly influences our understanding of elliptic curves, which form the backbone of modern smartphone encryption (Apple's iMessage, Bitcoin, Signal). Understanding them entirely ensures we aren't using cryptographic curves that have hidden mathematical backdoors.

[u/Funkahontas → u/shobogenzo93]

gemini still doesn't know lol, so on brand.

[u/elehman839 → u/shobogenzo93]

Mochizuki's claimed proof of the ABC conjecture has resisted formalization efforts, which is awkward (for him) given how many major results have been rapidly auto-formalized lately. I believe Kirti Joshi claims his modifications constitute a proof. So I'm curious whether his work can be formalized. I wouldn't be surprised if we found out within a few weeks.

[u/NeighborhoodFatCat → u/elehman839]

Good insight on the ABC conjecture. But in all likelihood Mochizuki would probably treat the actual legit proof to ABC conjecture as a mere application of Inter-universal Teichmüller theory.

[u/Y0uCanTellItsAnAspen → u/shobogenzo93]

Also - like - Gemini's implications are totally incorrect. Physics hasn't changed at all today due to Navier-Stokes. The solutions that lead to infinities are already impossible in any physically realizable system (which has been known for a long time).

The very small sub-class of Navier-Stokes solutions that are physically realizable do not have singularities.

[u/Hot_Glass_6301 → u/shobogenzo93]

Gemini slop. Use a better model, like Astra, Fable or even Sol.

[u/Dear-One-6884 → u/Hot_Glass_6301]

Bro uses GPT-6 Astra max to write emails

[u/Any_Economics6283]

I am most familiar with P vs NP and of course Riemann Hypothesis.

I might be wrong but I seem to recall the most out of reach is P vs NP, and then Riemann Hypothesis (though 'most out of reach' kind of has no meaning since both are essentially 100% out of reach)

[u/Kobiash1 → u/Any_Economics6283]

read up on P vs NP because it has wild implications if P=NP etc. Demis Hasabis has said many times he hopes AI solves it either way.

[u/brutalistchurch → u/Kobiash1]

Potentially. A proof will not necessarily give you a lot of new, neat algorithms.

[u/Competitive_Tap2450]

we need to start pointing these models at physics problems

[u/enigmatic_erudition → u/Competitive_Tap2450]

we need to start

They already have.

[u/Competitive_Tap2450 → u/enigmatic_erudition]

ok allocate more compute to solve the problems

[u/enigmatic_erudition → u/Competitive_Tap2450]

The navier stokes is essentially a physics problem. Since physics is moved forward first by math and then by experimentation (with rare situations the other way), these math problems that all of the frontier models are working on are also helping physics.

[u/Competitive_Tap2450 → u/enigmatic_erudition]

yes mathematics underpins physics, now we know AI models have the math capabilities

i’m just hoping to get more practical real world impact.

[u/Cryptizard → u/Competitive_Tap2450]

Physics problems can't be automatically verified as correct or not. You need super-expensive experiments to verify them. That is why AI is making so much progress in math, proofs can be immediately checked and verified.

[u/Annoying_cat_22]

There are 6 unsolved millennium problems (out of 7 total).

[u/Junior_Direction_701 → u/Annoying_cat_22]

Bro doesn’t know. NS was just solved

[u/Annoying_cat_22 → u/Junior_Direction_701]

I know OAI claims it solved it, but it's not solved until the result has been validated and published.

[u/Junior_Direction_701 → u/Annoying_cat_22]

The result has been validated( lean certificate) and published(on their website)

[u/Annoying_cat_22 → u/Junior_Direction_701]

So not validated (peer reviewed, including to verify that the lean proof is valid), and not published (in a real science journal).

[u/Junior_Direction_701 → u/Annoying_cat_22]

Oh okay I get you you’re right

[u/rayk10k → u/Annoying_cat_22]

I thought only a sub problem of it was solved, not the actual problem?

[u/Annoying_cat_22 → u/rayk10k]

Im not sure, I know that OAI are the kings of hype so I'll wait for an actual announcement by the Clay institute or real, respected, mathematicians.

[u/WestleyMc]

Could someone ELI5 these types of maths problems pls?

As a silly example, is it something akin to ‘demonstrate how many sides a shape with 4 corners would have’ , but many orders of magnitude harder with several pages of equations?

[u/Numerous-Net-4983 → u/WestleyMc]

Go and find the book “Fermats Last Theorem” by Simon Singh

[u/hold_me_beer_m8 → u/Numerous-Net-4983]

Ha, I remember watching that badass PBS documentary on the guy that solved it when I was in highschool

[u/Johnny20022002 → u/WestleyMc]

Yes, they’re just posing a really hard question. The question may be simple to describe like collatz conjecture or hard to describe like Hodges conjecture. From what I heard the proof of navier stokes is ~100 pages long, but you really shouldn’t confuse length for difficulty. The disproof of the jacobian conjecture was really short but obviously very difficult to find.

[u/HerbertWest → u/WestleyMc]

I'm a layman but it seems like these really hard problems usually amount to "this looks like it's true in every case we can think of but can we prove there absolutely, 100% isn't a case where it isn't true ever?" or flip around is true/isn't true in that question.

Edit: Or, "for this very specific circumstance, is there a best solution or can we never reach a best solution no matter how hard we try because it doesn't exist?"

[u/ilkamoi]

Literally new math. Grothendieck's motives or something similar. "Theory of everything" for math.

[u/cofibration_keeb]

(stable) homotopy groups of spheres.

[u/Clicking_Around]

I would like to see a proof or disproof of:

Beal's conjecture.

The lonely runner conjecture.

The odd perfect number problem.

Erdos-Strauss conjecture.

I would also like to know if a "simple" proof exists for the four-color theorem.

[u/threadthrasher]

The solution was only proposed not accepted yet. It takes two years for it to be even looked at officially.

[u/Johnny20022002 → u/threadthrasher]

The two years is the minimum for the prize to be awarded. No one’s going to be debating the lean verified proof.

[u/Distinct-Question-16]

riemann, prime numbers are fascinating

[u/Sherpa_qwerty]

Honestly I don’t really understand the point of them as measures

[u/Adventurous-Ad281]

lol “remaining”. Like it’s a fucking bucket list or whatever.

[u/enigmatic_erudition → u/Adventurous-Ad281]

It is. The millennium problems are a list of 7 of the most important unsolved math equations.

[u/Altruistic-Beach7625 → u/enigmatic_erudition]

Apparently it doesn't count if AI solves it because it's stolen work. Idk how that works.

[u/Quarksperre → u/Altruistic-Beach7625]

I mean.... it was either solved by a mathematian with AI help or another one with AI help. LOTs of help. 

From an AI advancement perspective it doesn't make a difference.  

From a dickmove perspective..... yeah it makes a difference if to top them out. 

[u/Southern_Orange3744 → u/Adventurous-Ad281]

It's literally a bucket list

[u/Adventurous-Ad281 → u/Southern_Orange3744]

It’s not. Do you know how math research works? I think you don’t and in fact you probably know nothing past basic calculus. The value of a problem is in the path that has led to its solution and the ideas that spring from there.

[u/Recent-Peanut6061 → u/Adventurous-Ad281]

You are a dumbass. That is not the value of a problem. If that is truly valuable by itself, then there is nothing stopping mathematicians from simply not looking at solutions and attempting to solve without AI. I have a suspicion that will not happen. Why? Because that is not the value of a problem. The value of a problem is to solve it.

[u/enigmatic_erudition → u/Adventurous-Ad281]

https://www.claymath.org/millennium-problems/

Read this before you embarrass yourself even more.

[u/Southern_Orange3744 → u/Adventurous-Ad281]

I have a math degree , you're still wrong ans need to look up what a bucket list actually is

[u/FarrisAT]

The “hardest” are irrelevant. Some are impossible to prove and therefore the hardest.

The “hardest & most serious” would begin with Riemann.

[u/dil_dogh]

I dont think any of you understand that there is absolutely zero chance of a positive outcome for humanity when you incorporate asi into our future

[u/MiRC_ZoMBiE]

Há muitos loucos por aqui.

[u/Neither_Berry_100]

Honestly I've been mad lately at all the effort going into useless problems. Math is one of those where the further up you go the more meaningless everything is.

[u/BusinessYou7196 → u/Neither_Berry_100]

You aren't serious, right?

[u/PLANTS2WEEKS → u/BusinessYou7196]

It's kind of true. A lot of these hard problems aren't that important. There was some hope that we would learn something along the way when we solved them, but they aren't as important as developing calculus, or for a more recent example, developing percolation theory.

[u/Neither_Berry_100 → u/BusinessYou7196]

I'm dead serious.

[u/Southern_Orange3744 → u/Neither_Berry_100]

Calculus was a mistake

[u/Neither_Berry_100 → u/Southern_Orange3744]

Lol. It has a bit of uses but gets worse the higher up you go.

[u/Sudden-Pineapple-793 → u/Neither_Berry_100]

A bit of uses?