Reference hub
Unresolved Mathematical Mysteries: What Is Still Open?
Maintained by Sudershan Soni Status checked: 2 October 2026
A curated selection of 15 major unsolved (and a few famously claimed, disputed or historically solved) mathematical problems — not an exhaustive list of every open question in mathematics. Every entry here is held to the same standard: what's actually proven, what's merely suspected, and what the difference means. Further down, Solved — And How covers 11 problems that were eventually settled — by hand, by computer, and now by AI.
Five words that mean five different things
Conjecture
A precise mathematical statement believed likely true, based on evidence or pattern, but not yet proven.
Proof
A logical argument that establishes a statement is true in every single case, with no exceptions possible — not just the cases checked.
Counterexample
A single genuine case where a conjecture fails — enough on its own to disprove it completely, however many cases supported it before.
Numerical evidence
Confirmation that a statement holds for every case actually tested — however large that number of cases, it is never the same as a proof covering every case.
Independence
A proof that a statement can be neither proved nor disproved from a given set of starting rules (axioms) — a different, final kind of answer, not an open question awaiting cleverness.
Checking examples vs. proving it for every case
Drag the slider all the way to the right. However far you push it, the bar never actually reaches “every whole number” — there are infinitely many, and no amount of checking, by hand or by computer, can finish. That's the real difference between strong numerical evidence (like 2^68 cases confirmed for Collatz) and a mathematical proof, which covers every case at once through pure logic, with no checking left to do.
Keep this in mind for every 'Open' entry below: no amount of checking finishes the job the way a proof does.
Overview
| Problem | Field | Status |
|---|---|---|
| The Riemann Hypothesis | Number theory | Open |
| P versus NP | Computer science / mathematical logic | Open |
| Navier–Stokes Existence and Smoothness | Fluid dynamics / partial differential equations | Claimed resolution — under evaluation |
| Yang–Mills Existence and Mass Gap | Mathematical physics | Open |
| The Hodge Conjecture | Algebraic geometry | Open |
| Birch and Swinnerton-Dyer Conjecture | Number theory | Open |
| The Poincaré Conjecture | Topology | Solved historical problem |
| Twin Prime Conjecture | Number theory | Open |
| Strong Goldbach Conjecture | Number theory | Open |
| Collatz Conjecture (3n + 1 problem) | Number theory / dynamical systems | Open |
| Do Odd Perfect Numbers Exist? | Number theory | Open |
| The abc Conjecture | Number theory | Disputed proof |
| The Beal Conjecture | Number theory | Open |
| Generalized Riemann Hypothesis (GRH) | Number theory | Open |
| The Continuum Hypothesis | Set theory / mathematical logic | Independent of specified axioms |
Showing 15 of 15 problems.
OpenThe Riemann Hypothesis
Number theory
Do all the 'non-trivial zeros' of the Riemann zeta function lie exactly on one specific line?
Full details
The Riemann Hypothesis
Number theory
Do all the 'non-trivial zeros' of the Riemann zeta function lie exactly on one specific line?
The Riemann zeta function is a mathematical object built from an infinite sum that, remarkably, encodes deep information about how prime numbers are spread out. It equals zero at certain special points; the 'trivial' ones are well understood, but infinitely many 'non-trivial' ones aren't. The hypothesis predicts every single one sits on a precise vertical line (where the real part of the input equals 1/2). Over 10 trillion have been checked by computer — every one is exactly on the line.
Precise statement (deeper level)
All non-trivial zeros of ζ(s) have real part equal to 1/2.
Why it matters
An enormous body of existing mathematics is built on the assumption that RH is true — hundreds of published theorems are stated as 'true, if RH is true.' A proof would instantly promote all of them to unconditional facts, and would pin down the error term in the Prime Number Theorem to its sharpest possible form.
What is known
Over 10 trillion zeros verified computationally, all on the line. A positive proportion of all zeros (over 40%) have been proven to lie on the line using analytic techniques, without relying on computation.
What remains unresolved
A proof (or disproof) covering every one of the infinitely many zeros, not just those checked or covered by current partial results.
Verified progress, with dates
- 1859Bernhard Riemann states the hypothesis in a paper on the distribution of primes.
- 2004-2020sComputational verification extended past 10 trillion zeros with no exception found.
Does this progress settle the question?
Overwhelming numerical and partial analytic evidence, but no proof exists for the infinite remainder — this is the textbook example of evidence that is not a proof.
Sources
Status checked: 2 October 2026.
OpenP versus NP
Computer science / mathematical logic
Can every problem whose solution is quick to check also always be solved quickly, not just verified?
Full details
P versus NP
Computer science / mathematical logic
Can every problem whose solution is quick to check also always be solved quickly, not just verified?
Some problems — like a finished sudoku, or a proposed delivery route under a target length — take almost no time to check once you have a candidate answer. P vs NP asks whether that whole broad class of 'quick to check' problems can always also be solved quickly from scratch, as a matter of pure logic, or whether checking and solving are fundamentally different difficulties.
Precise statement (deeper level)
Is the complexity class P (solvable in polynomial time) equal to the complexity class NP (solution verifiable in polynomial time)?
Why it matters
Much of modern cryptography (including the encryption protecting online banking) relies on certain problems being easy to check but hard to solve. If P = NP were proved true, with an actual efficient method found, most current encryption would need replacing.
What is known
P is a subset of NP (anything quickly solvable is also quickly checkable) — that direction is trivial and proven. Whether the reverse holds is unknown. No problem has been proven to require more than polynomial time to solve while staying easy to check.
What remains unresolved
A proof either that P = NP (and ideally a genuinely efficient method for an NP problem) or that P ≠ NP (that no such method can ever exist for at least one NP-complete problem).
Verified progress, with dates
- 1971Stephen Cook formalises the P vs NP question and NP-completeness.
- OngoingDecades of attempted proofs on both sides; several well-known proof strategies have been shown NOT to work (so-called 'barrier' results), narrowing the search.
Does this progress settle the question?
Barrier results tell mathematicians which proof techniques definitely cannot work, which is real progress — but they don't themselves prove the answer either way.
Sources
Status checked: 2 October 2026.
OpenYang–Mills Existence and Mass Gap
Mathematical physics
Can the equations behind the Standard Model of particle physics be made fully mathematically rigorous, and do they genuinely force a 'mass gap' to exist?
Full details
Yang–Mills Existence and Mass Gap
Mathematical physics
Can the equations behind the Standard Model of particle physics be made fully mathematically rigorous, and do they genuinely force a 'mass gap' to exist?
Yang–Mills theory underlies how fundamental forces work in particle physics, and physicists use it with huge experimental success. The missing piece is a fully rigorous mathematical construction of the theory (not just a physically successful approximation), plus a proof that it necessarily predicts a 'mass gap' — that the lightest particle the theory allows for must have strictly positive mass, matching what experiments actually observe.
Why it matters
It sits exactly at the junction of physics and pure mathematics; solving it would be one of the deepest confirmations that the maths underneath modern physics is logically airtight, not merely empirically successful.
What is known
Yang–Mills theory works extremely well as physics, confirmed to extraordinary precision by particle accelerator experiments. A rigorous mathematical version exists in some simplified (lower-dimensional) settings.
What remains unresolved
A fully rigorous construction of the full 4-dimensional quantum Yang–Mills theory, plus a proof that a mass gap is a necessary consequence of that construction.
Verified progress, with dates
- 1954Chen Ning Yang and Robert Mills introduce the theory.
- OngoingRigorous constructions exist in simplified/lower-dimensional cases; the full physical case remains open.
Does this progress settle the question?
Considered among the most technically demanding of the seven Millennium Problems; real progress exists in restricted settings, but it does not settle the general case.
Sources
Status checked: 2 October 2026.
OpenThe Hodge Conjecture
Algebraic geometry
For a well-behaved family of shapes built from polynomial equations, does every 'natural' internal structure actually come from simpler algebraic pieces?
Full details
The Hodge Conjecture
Algebraic geometry
For a well-behaved family of shapes built from polynomial equations, does every 'natural' internal structure actually come from simpler algebraic pieces?
Algebraic geometry studies shapes defined by polynomial equations. Such shapes have internal features describable using topology (shape in a flexible, stretch-and-bend sense). The Hodge Conjecture predicts that, for a specific well-behaved class of these shapes, every topological feature of this kind can also be built directly from algebraic sub-shapes (pieces that are themselves solutions to polynomial equations) — essentially, that two different lenses for looking at the same shape must always agree.
Why it matters
It's a fundamental question about how two major branches of geometry — topology and algebra — relate to each other; a proof would tie them together far more tightly than currently known.
What is known
Proven true in many specific, lower-dimensional or symmetric cases.
What remains unresolved
A general proof, or a counterexample, covering the full class of shapes the conjecture applies to.
Verified progress, with dates
- 1950William Hodge presents the conjecture.
- OngoingVerified in numerous special cases; no general proof or counterexample found.
Does this progress settle the question?
Widely regarded as one of the hardest to state precisely for a general audience, let alone prove — special-case progress has not produced a path to the general statement.
Sources
Status checked: 2 October 2026.
OpenBirch and Swinnerton-Dyer Conjecture
Number theory
Can you predict how many rational-number solutions an elliptic curve has, just from the behaviour of a related function?
Full details
Birch and Swinnerton-Dyer Conjecture
Number theory
Can you predict how many rational-number solutions an elliptic curve has, just from the behaviour of a related function?
An elliptic curve is an equation of the form y² = x³ + ax + b — central to modern cryptography and to Andrew Wiles' proof of Fermat's Last Theorem. Some elliptic curves have infinitely many rational solutions; others only finitely many. The conjecture says you can tell which — and estimate how 'rich' the solutions are — purely from the behaviour, near one specific point, of a separate function built from the curve (its L-function).
Why it matters
It would replace a currently case-by-case, often computationally hard question ('how many solutions does this curve have?') with a clean, provable rule, with direct relevance to cryptography and number theory.
What is known
Verified computationally for a very large number of individual elliptic curves — proven in some restricted general cases (e.g. curves of 'rank' 0 or 1).
What remains unresolved
A general proof covering elliptic curves of every rank.
Verified progress, with dates
- 1960sBryan Birch and Peter Swinnerton-Dyer formulate the conjecture from computational experiments.
- 1980s-2020sProven for curves of rank 0 and 1 by various mathematicians; remains open in general.
Does this progress settle the question?
Real partial proofs exist for specific, restricted cases, which is genuine progress, but they don't extend to a general argument covering every elliptic curve.
Sources
- Clay Mathematics Institute — Birch and Swinnerton-Dyer Conjecture
- Birch and Swinnerton-Dyer conjecture — Wikipedia
Status checked: 2 October 2026.
Solved historical problemThe Poincaré Conjecture
Topology
If a 3-dimensional shape has no holes in a precise topological sense, must it secretly be a (higher-dimensional) sphere?
Full details
The Poincaré Conjecture
Topology
If a 3-dimensional shape has no holes in a precise topological sense, must it secretly be a (higher-dimensional) sphere?
In two dimensions this is almost obvious: any loop on a sphere's surface can shrink to a point without leaving the surface; a loop around a doughnut's hole cannot. The conjecture asked whether the same 'no holes means it's secretly a sphere' rule holds one dimension up, for 3-dimensional shapes in 4-dimensional space.
Why it matters
It was the last and hardest of Henri Poincaré's foundational questions in topology to be resolved, and the technique used to solve it (Ricci flow) has become a major tool across geometry.
What is known
Proven true. Grigori Perelman's 2002-2003 papers, building on Richard Hamilton's Ricci flow programme, provided a complete proof, independently verified by multiple teams by around 2006.
What remains unresolved
Nothing — this is the one Millennium Prize Problem that is genuinely solved.
Verified progress, with dates
- 1904Henri Poincaré poses the conjecture.
- 1982-2003Richard Hamilton develops Ricci flow as a tool; Grigori Perelman completes the proof using it.
- ~2006Independent verification teams confirm the proof is correct.
- 2006Perelman is awarded, and declines, the Fields Medal.
- 2010The Clay Institute formally awards, and Perelman declines, the $1 million Millennium Prize.
Does this progress settle the question?
Fully proven and independently verified — the clearest possible example of the difference between a claim and an accepted proof, since this one genuinely completed every stage.
Sources
Status checked: 2 October 2026.
OpenTwin Prime Conjecture
Number theory
Are there infinitely many pairs of primes that differ by exactly 2, like (11, 13) or (17, 19)?
Full details
Twin Prime Conjecture
Number theory
Are there infinitely many pairs of primes that differ by exactly 2, like (11, 13) or (17, 19)?
Prime pairs like (11, 13) get rarer as numbers grow, but they keep turning up — (1,000,000,000,061, 1,000,000,000,063) is one example far out. The conjecture says this never truly stops, however large the numbers get.
Why it matters
It's one of the oldest, simplest-to-state open questions about the primes, and progress on it has driven major new sieve techniques used across number theory.
What is known
In 2013, Yitang Zhang proved there are infinitely many prime pairs with SOME bounded gap (originally under 70 million) — not necessarily exactly 2, but a fixed, finite gap. Subsequent work (notably by James Maynard and Terence Tao) rapidly shrank that bound to 246.
What remains unresolved
Shrinking the proven gap from 246 all the way down to exactly 2 — a gap that current sieve techniques are not believed capable of closing without a genuinely new idea.
Verified progress, with dates
- 2013Yitang Zhang proves a bounded prime gap exists infinitely often (under 70,000,000).
- 2013-14The Polymath8 project and James Maynard independently shrink the bound to 246.
Does this progress settle the question?
Genuine, celebrated mathematical progress — proving SOME bounded gap exists was a historic breakthrough — but 246 is not 2, and nobody currently has a technique believed capable of closing that remaining distance.
Sources
Status checked: 2 October 2026.
OpenStrong Goldbach Conjecture
Number theory
Can every even whole number greater than 2 be written as the sum of two primes?
Full details
Strong Goldbach Conjecture
Number theory
Can every even whole number greater than 2 be written as the sum of two primes?
100 = 3 + 97, or 11 + 89, or several other pairs; 10 = 3 + 7 or 5 + 5. The conjecture says every even number keeps this property, however large it gets.
Why it matters
One of the oldest unsolved problems in all of mathematics (first proposed 1742), and a natural testing ground for understanding how additive structure and prime numbers interact.
What is known
Verified by computer for every even number up to about 4 × 10^18. A weaker, related statement — every odd number greater than 5 is the sum of three primes ('weak' or ternary Goldbach) — was fully proven by Harald Helfgott in 2013.
What remains unresolved
A general proof of the two-prime (strong) version for every even number, not just those checked by computer. Helfgott's 2013 result, though a genuine landmark, proves a different (three-prime, odd-number) statement and does not establish the two-prime version.
Verified progress, with dates
- 1742Christian Goldbach proposes the conjecture in a letter to Euler.
- 2013Harald Helfgott proves the weaker 'ternary' (three-prime, odd numbers) version.
- 2000s-2020sComputer verification of the strong (two-prime) version extended past 4 × 10^18 with no exception.
Does this progress settle the question?
Helfgott's result is a real, fully accepted proof — but of a genuinely different statement. It's frequently, and incorrectly, cited as if it solved the (still open) two-prime version.
OpenCollatz Conjecture (3n + 1 problem)
Number theory / dynamical systems
Starting from any whole number, if you repeatedly halve even numbers and triple-plus-one odd numbers, do you always eventually reach 1?
Full details
Collatz Conjecture (3n + 1 problem)
Number theory / dynamical systems
Starting from any whole number, if you repeatedly halve even numbers and triple-plus-one odd numbers, do you always eventually reach 1?
Take any whole number. If it's even, halve it. If it's odd, triple it and add 1. Repeat. Every number ever tried this way — however it wanders up and down first — eventually lands on 1. Try the explorer on our Millennium Prize Problems article to see this for yourself.
Why it matters
Famous for being extraordinarily simple to state and extraordinarily resistant to proof — mathematician Paul Erdős reportedly said mathematics was 'not yet ready' for problems like this one.
What is known
Verified by computer for every starting number up to roughly 2^68 (about 300 quintillion) — all reach 1. In 2019, Terence Tao proved that 'almost all' starting numbers (in a precise statistical sense) eventually reach a very small value, a major partial result.
What remains unresolved
A proof that EVERY starting number, with no exception whatsoever, reaches 1 — Tao's 2019 result, while a genuine landmark, explicitly does not rule out rare exceptions.
Verified progress, with dates
- 1937Lothar Collatz proposes the conjecture.
- 2019Terence Tao proves that almost all orbits attain almost bounded values — a major, but not complete, result.
Does this progress settle the question?
Tao's result is genuinely significant progress, repeatedly and accurately described by Tao himself as falling short of a full proof — 'almost all' is a precise technical term that still permits exceptions.
OpenDo Odd Perfect Numbers Exist?
Number theory
Is there an odd whole number whose proper divisors (not counting itself) add up to the number itself?
Full details
Do Odd Perfect Numbers Exist?
Number theory
Is there an odd whole number whose proper divisors (not counting itself) add up to the number itself?
6 is 'perfect': its divisors 1, 2 and 3 add up to 6. All known perfect numbers (6, 28, 496, 8128…) are even. Nobody has ever found an odd one, and nobody has proven one can't exist.
Why it matters
A simply-stated existence question that has resisted proof for over two thousand years, and a useful testing ground for techniques in multiplicative number theory.
What is known
Any odd perfect number, if one exists, has been proven to need an extremely restrictive structure: it must be larger than 10^1500, have at least 10 distinct prime factors, and satisfy many other narrow conditions.
What remains unresolved
Either a proof that no odd perfect number can exist under any circumstances, or an actual example (expected, if one exists at all, to be astronomically large).
Verified progress, with dates
- Ongoing, 20th-21st centuryA long sequence of results progressively narrows the possible structure any odd perfect number would have to have, without ruling out existence entirely.
Does this progress settle the question?
Decades of work have made the space of possibilities extremely narrow without closing it off completely — a case where constraints have piled up without yet forcing a contradiction.
Disputed proofThe abc Conjecture
Number theory
For three whole numbers a + b = c with no shared factors, is there a precise limit on how 'small' the prime factors involved in a, b and c can collectively be?
Full details
The abc Conjecture
Number theory
For three whole numbers a + b = c with no shared factors, is there a precise limit on how 'small' the prime factors involved in a, b and c can collectively be?
If a and b share no common factors and a + b = c, the conjecture relates the size of c to the product of the distinct prime numbers that divide a, b and c together (called the 'radical'). Loosely: c can't be much bigger than that radical, except in a controlled, rare way. It's a deep statement about how addition and multiplication of whole numbers interact.
Why it matters
If proven, it would imply, in a single stroke, many other major results and conjectures in number theory, including a strong form of Fermat's Last Theorem for all but finitely many exponents.
What is known
Shinichi Mochizuki announced a purported proof in 2012, using a new, highly technical framework he calls Inter-universal Teichmüller (IUT) theory.
What remains unresolved
Broad acceptance of a correct proof. Mochizuki's claimed proof has not been accepted by most of the specialist mathematical community.
Verified progress, with dates
- 2012Mochizuki posts four papers claiming a proof via IUT theory.
- 2018Mathematicians Peter Scholze and Jakob Stix publish a detailed report identifying what they describe as a serious, unresolved gap in the argument.
- 2021The papers are published in RIMS (a journal Mochizuki edits), without resolving the Scholze–Stix objections to the satisfaction of most outside experts.
Does this progress settle the question?
This is the clearest example in modern mathematics of the gap between 'a proof has been published' and 'a proof is broadly accepted' — publication in a journal did not settle the disagreement among specialists, which remains genuinely unresolved.
Sources
- abc conjecture — Wikipedia
- Why abc is still a conjecture — Peter Scholze & Jakob Stix report (via Scholze's site)
Status checked: 2 October 2026.
OpenThe Beal Conjecture
Number theory
If Aᵖ + Bᵍ = Cʳ (with p, q, r all greater than 2), must A, B and C share a common prime factor?
Full details
The Beal Conjecture
Number theory
If Aᵖ + Bᵍ = Cʳ (with p, q, r all greater than 2), must A, B and C share a common prime factor?
This generalises Fermat's Last Theorem (which is the special case p = q = r). The conjecture says that whenever such an equation has a solution with all exponents above 2, the three base numbers A, B, C cannot be completely coprime — they must share at least one prime factor.
Why it matters
It's a natural extension of one of the most famous solved problems in mathematics (Fermat's Last Theorem), asking what happens when the exponents are allowed to differ.
What is known
No counterexample has ever been found; the conjecture has been numerically verified across an enormous search space. Businessman and mathematics enthusiast Andrew Beal has personally offered a $1 million prize for a proof or counterexample, administered by the American Mathematical Society.
What remains unresolved
A general proof, or a counterexample.
Verified progress, with dates
- 1993-97Andrew Beal formulates and popularises the conjecture, offering a prize.
- OngoingThe prize fund has grown over time (reported at $1 million); no proof or counterexample has been accepted.
Does this progress settle the question?
An extensive, still-growing computer search has found zero counterexamples, which is suggestive but — exactly like the other numerically-tested conjectures on this page — not a proof.
OpenGeneralized Riemann Hypothesis (GRH)
Number theory
Does the same 'all zeros on one line' pattern hold not just for the Riemann zeta function, but for a much wider family of related functions (Dirichlet L-functions)?
Full details
Generalized Riemann Hypothesis (GRH)
Number theory
Does the same 'all zeros on one line' pattern hold not just for the Riemann zeta function, but for a much wider family of related functions (Dirichlet L-functions)?
The ordinary Riemann Hypothesis (above) is about one specific function. Mathematicians also study a whole family of closely related functions, each tied to patterns in arithmetic progressions (numbers like 3, 7, 11, 15… that increase by a fixed step). GRH extends the same 'zeros on one line' prediction to every function in that entire family at once.
Why it matters
A great deal of modern number theory — particularly results about primes in arithmetic progressions, and several algorithms in computational number theory — is proven conditionally on GRH; a proof would make all of it unconditional.
What is known
Proving GRH would automatically prove the ordinary Riemann Hypothesis as a special case — so GRH is, in a precise sense, an even harder, strictly stronger open question.
What remains unresolved
Everything the ordinary Riemann Hypothesis still needs, extended across infinitely many additional functions at once.
Verified progress, with dates
- OngoingShares the same computational-verification and partial-analytic-result pattern as the ordinary Riemann Hypothesis, extended across more functions, with correspondingly less covered so far.
Does this progress settle the question?
Since GRH implies RH, and RH itself remains unproven, GRH cannot currently be closer to resolved than RH is — it inherits the same open status, for a strictly larger problem.
Independent of specified axiomsThe Continuum Hypothesis
Set theory / mathematical logic
Is there a size of infinity strictly between the size of the whole numbers and the size of the real numbers?
Full details
The Continuum Hypothesis
Set theory / mathematical logic
Is there a size of infinity strictly between the size of the whole numbers and the size of the real numbers?
There is more than one 'size' of infinity — the whole numbers (1, 2, 3…) are infinite, but the real numbers (every point on a number line, including all the decimals in between) form a strictly bigger infinity. The Continuum Hypothesis asks whether there's a size of infinity strictly in between those two, or not.
Why it matters
It was the first problem on David Hilbert's famous 1900 list of 23 open problems, and the story of its resolution fundamentally changed what mathematicians understand a 'proof' to even be capable of settling.
What is known
This is not an ordinary open conjecture waiting for cleverness to crack it. Kurt Gödel proved in 1940 that the Continuum Hypothesis CANNOT be disproved from the standard axioms of set theory (ZFC). Paul Cohen then proved in 1963 that it also CANNOT be proved true from those same axioms. Together, these results mean the Continuum Hypothesis is independent of ZFC: the standard rules of mathematics are simply silent on the question, in either direction.
What remains unresolved
Nothing remains to be proven within standard ZFC set theory — that question is closed, permanently, by Gödel and Cohen's results. What remains is a genuinely different, ongoing question: whether some new, independently well-motivated axiom ought to be added to set theory that would settle it one way or the other — an active, but philosophically different, area of research.
Verified progress, with dates
- 1878Georg Cantor poses the Continuum Hypothesis.
- 1900David Hilbert lists it as the first of 23 major open problems.
- 1940Kurt Gödel proves CH cannot be disproved from ZFC.
- 1963Paul Cohen proves CH also cannot be proved from ZFC, establishing independence.
Does this progress settle the question?
Independence is itself the final, complete answer within ZFC — this is categorically different from every other entry on this page, where 'open' means 'not yet proven' rather than 'proven unprovable.'
Sources
Status checked: 2 October 2026.
Once open, now settled
Solved — And How
Every problem below was once exactly where the open ones above are now. Some fell to pure human reasoning, some needed a computer to check more cases than any person could, and since 2025 a new group has appeared: results where AI found the key idea itself. For each one we say plainly what the machine actually did and how the result has been checked. Status checked: 3 October 2026.
Fermat's Last Theorem
Posed 1637 · Solved 1994 (published 1995)
Andrew Wiles, with Richard Taylor
Can aⁿ + bⁿ = cⁿ ever be true in whole numbers when n is bigger than 2?
Pierre de Fermat scribbled in a book margin that he had a marvellous proof that the answer is no — and that the margin was too small to hold it. For 358 years nobody found one. Andrew Wiles worked on it largely in secret for seven years, announced a proof in 1993, then found a gap in it; with his former student Richard Taylor he repaired it in 1994.
What the computer or AI did
None. This is the classic modern example of a deep result proved entirely by human reasoning.
How we know it's right
Published in the Annals of Mathematics in 1995 after expert refereeing; universally accepted.
The Four Colour Theorem
Posed 1852 · Solved 1976
Kenneth Appel and Wolfgang Haken
Can every flat map be coloured with only four colours so that no two neighbouring countries share a colour?
Francis Guthrie noticed in 1852 that four colours always seemed enough for a map of England's counties. Appel and Haken reduced the whole problem to checking almost two thousand special map configurations — far too many to check by hand — and used over a thousand hours of computer time to check them. It was the first famous theorem whose proof no human could read through in full.
What the computer or AI did
Checked every one of the configurations the human argument reduced the problem to.
How we know it's right
Controversial at first because nobody could verify it by hand. A simpler computer proof followed in 1996 (Robertson, Sanders, Seymour and Thomas), and in 2005 Georges Gonthier and Benjamin Werner checked the entire proof inside the Coq proof assistant — every logical step verified by machine.
Illustration — why four colours can be necessary
The Robbins Conjecture
Posed 1930s · Solved 1996
William McCune's automated prover EQP (Argonne National Laboratory)
Do three simple rules (Robbins' axioms) describe exactly the same algebra as standard Boolean algebra — the logic inside every computer chip?
Herbert Robbins proposed the question soon after 1933, and some of the best logicians of the century, including Alfred Tarski, tried and failed. In 1996 William McCune's automated theorem prover, EQP, searched for a proof on its own and found one after about eight days of computation.
What the computer or AI did
Found the proof itself — not just checking cases a human had set up. One of the first genuinely open problems settled this way.
How we know it's right
The proof was short enough for humans to read and check, and it was confirmed correct.
The Kepler Conjecture
Posed 1611 · Solved 1998 (formally verified 2014)
Thomas Hales, with Samuel Ferguson
Is the greengrocer's pyramid of oranges the densest possible way to pack identical spheres?
Johannes Kepler claimed in 1611 that no packing of equal spheres beats the familiar stacking, which fills about 74% of space. Thomas Hales turned the problem into a huge optimisation over thousands of possible arrangements and used computers to solve it, announcing a proof in 1998. The referees reported they were '99% certain' it was right but could not check every computation.
What the computer or AI did
Solved thousands of optimisation problems that together rule out every denser arrangement.
How we know it's right
To remove the remaining doubt, Hales led the Flyspeck project, which checked the whole proof in proof-assistant software. It was completed in 2014, so the result is now fully machine-verified.
Illustration — the greengrocer's stack
Density of this arrangement: π / √18 ≈ 74.05% of space filled
The Boolean Pythagorean Triples Problem
Posed 1980s · Solved 2016
Marijn Heule, Oliver Kullmann and Victor Marek
Can the whole numbers be coloured red or blue so that no Pythagorean triple (like 3, 4, 5) is all one colour?
Ronald Graham offered $100 for the answer. A 'SAT solver' — software for huge true/false logic puzzles — ran for about two days on the Stampede supercomputer in Texas. The answer: you can do it for every number from 1 to 7,824, but at 7,825 it becomes impossible.
What the computer or AI did
Searched about a trillion cases, cleverly narrowed down, and produced a certificate that every one fails.
How we know it's right
The certificate was 200 terabytes — at the time reported by Nature as the largest maths proof ever — and was checked by independent proof-checking software. Heule received Graham's $100 prize.
Try it yourself — colour 1 to 40 with no one-colour Pythagorean triple
Tap a number to switch it between blue and red. Goal: no triple a² + b² = c² (like 3, 4, 5) may be all one colour. Numbers in a clashing triple get a dark ring.
16 one-colour triples: 3–4–5, 5–12–13, 6–8–10, 7–24–25…
Keller's Conjecture (dimension 7)
Posed 1930 · Solved 2019
Joshua Brakensiek, Marijn Heule, John Mackey and David Narváez
If you fill space with identical cubes, must at least two of them share a whole face?
Ott-Heinrich Keller asked this for every dimension. It was proved true up to dimension 6 and shown false from dimension 8 upward — leaving dimension 7 as the last gap for decades. In 2019 a team turned the question into a SAT problem and showed the conjecture is true in seven dimensions.
What the computer or AI did
Settled the final remaining case by exhaustive, symmetry-reduced search.
How we know it's right
Produced a roughly 200-gigabyte proof certificate, checked by independent proof-checking software.
The Kakeya Conjecture in three dimensions
Posed 1917 (needle problem); conjecture in its modern form later · Solved 2025
Hong Wang and Joshua Zahl
How 'big' must a set be if a needle can point in every direction inside it?
Sōichi Kakeya asked how little room you need to turn a needle all the way round. In three dimensions the modern question is about dimension rather than area: must such a set be fully three-dimensional? Hong Wang and Joshua Zahl posted a 127-page proof in February 2025 that the answer is yes. It was a major reason Wang received a 2026 Fields Medal.
What the computer or AI did
None — a human proof using multiscale analysis.
How we know it's right
Widely examined by experts since 2025 (including a detailed exposition by Terence Tao); journal review is the normal next step for a proof of this length.
Illustration — turning a needle all the way round
Graham's Rearrangement Conjecture
Posed 1971 · Solved 2026 (final paper February 2026)
Four papers: Müyesser & Pokrovskiy; Kravitz & Bedert; a joint team; and finally Lisa Sauermann & Huy Tuan Pham
Can any set of non-zero numbers on a prime-sized clock always be ordered so every running total is different?
Ronald Graham — mathematician and keen juggler — posed it in 1971. Different teams handled very small sets, very large sets and then more cases; the hardest middle range was closed by Sauermann and Pham in a 27-page paper in February 2026, using randomness and Fourier analysis.
What the computer or AI did
None — a human proof built on probability.
How we know it's right
Spread across four papers by different teams; reported by Quanta Magazine in September 2026 as officially closing the problem.
Try it yourself — distinct partial sums on an 11-hour clock
The numbers 1, 2, 3, 5, 9, 10, added up one at a time on a clock with 11 hours (so 12 o'clock is 0, and 13 wraps round to 2). Can you order them so every running total lands on a different hour?
| Order | 1 | 10 | 2 | 9 | 3 | 5 |
|---|---|---|---|---|---|---|
| Running total | 1 | 0 | 2 | 0 | 3 | 8 |
Repeated total: 0 — try again.
Erdős's Unit Distance Conjecture — disproved
Posed 1946 · Solved May 2026
An internal OpenAI reasoning model; checked by Noga Alon, Thomas Bloom, Tim Gowers and others
Among n points in a plane, how many pairs can be exactly 1 unit apart — barely more than n, as Erdős believed?
Paul Erdős conjectured that the number of unit-distance pairs can only grow a tiny bit faster than the number of points. In May 2026 an OpenAI model built an infinite family of point sets that beats that bound by a genuine power of n, using deep tools from algebraic number theory. Will Sawin then made the gain explicit: more than n to the power 1.014 pairs, later improved further.
What the computer or AI did
Found the key construction and the argument. OpenAI described it as the first time an AI had autonomously settled a central open problem in a branch of mathematics.
How we know it's right
Nine leading mathematicians, including Fields Medallists Tim Gowers and Jacob Tsimerman, checked it and wrote a companion paper. Gil Kalai compared its significance to the 1976 Four Colour proof.
Illustration — counting unit distances
19 points, 42 pairs exactly 1 unit apart
Nine Erdős problems — AlphaProof Nexus
Posed Various, up to 56 years old · Solved May 2026
Google DeepMind's AlphaProof Nexus (Gemini + the Lean proof checker)
Can an AI system produce machine-checked proofs of genuinely open research problems?
DeepMind pointed its AlphaProof Nexus agent at 353 open problems from the Erdős problems database. It solved 9 — including two that had been open for 56 years — and also proved 44 conjectures from the Online Encyclopedia of Integer Sequences.
What the computer or AI did
Searched for proofs and wrote them directly in Lean, a language a computer can check line by line.
How we know it's right
Every proof compiles in Lean, so each logical step is machine-verified; proofs are public on GitHub. Note the hit rate: 9 of 353, about 2.5%.
A non-sofic group exists
Posed 1999 · Solved August 2026 (announced)
OpenAI's unreleased model 'Astra'
Is every group 'sofic' — approximable by finite permutations, in the sense Mikhail Gromov introduced?
On 1 August 2026 OpenAI announced ten results from its next model, Astra, headlined by the first construction of a non-sofic group — answering a 27-year-old question in group theory — plus three Erdős problems and a first improvement since 1978 to a general sphere-packing bound.
What the computer or AI did
Produced the proofs and Lean certificates for all ten results.
How we know it's right
Lean certificates are public with no unproven steps. An independent human audit (Sienicki & Sienicki, final version 9 Sept 2026) found no confirmed substantive errors in the main results, though review depth varied and one result drew a specialist request for major revision. Mathematicians are already building on the group-theory result. Journal peer review had not concluded at our status-check date.
Read AI headlines carefully
Not every "AI solved it" headline is true
In October 2025, posts from OpenAI staff claimed GPT-5 had solved ten 'open' Erdős problems. Thomas Bloom, who runs the Erdős problems database, pointed out that 'open' on his site only meant he didn't know of a solution — the model had found existing solutions in old papers. The posts were deleted. Finding forgotten literature is genuinely useful, but it is not solving a problem.
The Decoder — OpenAI researcher announced a GPT-5 math breakthrough that never happenedThe full story of how machines went from checking cases in 1976 to finding proofs in 2026 is in our article Solved by Computer: Famous Maths Problems Machines Helped Crack.
Try two of these yourself
The Collatz Conjecture and the Goldbach Conjecture are both easy to test by hand — and both show exactly why testing isn't proving.
Try it yourself — Collatz explorer
Pick a whole number from 1 to 9,999. The rule: if it's even, halve it; if it's odd, triple it and add 1. Repeat.
27 → 82 → 41 → 124 → 62 → 31 → 94 → 47 → 142 → 71 → 214 → 107 → 322 → 161 → 484 → 242 → 121 → 364 → 182 → 91 … → 70 → 35 → 106 → 53 → 160 → 80 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1
111 steps, peaking at 9,232, landing on 1.
Every whole number ever tested this way — and computers have tested every one up to roughly 268, about 300 quintillion — eventually reaches 1. Nobody has proven it happens for every whole number, and nobody has found a counterexample. That gap between “true for every number we've checked” and “true for every number” is exactly what makes this an open problem, not a solved one.
Collatz Conjecture: every number tried reaches 1 — but 'every number tried' is not 'every number'.
Try it yourself — Goldbach pair finder
Pick an even number up to 100,000. The (strong) Goldbach conjecture says it can always be written as two primes added together.
100 =
- 3 + 97 = 100
- 11 + 89 = 100
- 17 + 83 = 100
- 29 + 71 = 100
- 41 + 59 = 100
- 47 + 53 = 100
Every even number ever checked — up to around 4 × 1018 by distributed computer searches — has at least one such pair. No exception has ever been found, and no proof exists that one never will be. (A weaker, related claim — every odd number above 5 is the sum of three primes — was proved by Harald Helfgott in 2013. That is a genuinely different, solved statement; it does not prove the stronger two-prime version above.)
Strong Goldbach Conjecture: every even number tried has a prime pair — but, again, not every even number has been tried.
Spotted an error, or know of a genuine update?
This page is reviewed and corrected by hand — it is not automatically updated or "live." If a status above is out of date or you believe something here is inaccurate, let us know and we'll check it.
Update history
- 3 October 2026 — Added “Solved — And How”: 11 problems that were once open, with how each was settled (by hand, computer-assisted, computer-found or AI), how each has been checked, and sources.
- 2 October 2026 — Page created; all 15 entries researched and status-checked against primary sources.
Want the seven official Millennium Prize Problems explained as one connected story, with worked examples? Read our companion article. Curious whether AI could solve any of these? Can AI Actually Do Maths?
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