Mathematics Atlas
Things this atlas does not know. Each entry is a question that specialists
have posed and not settled, recorded with what would actually resolve it and which
expertise the answer needs.
These are kept deliberately, rather than quietly omitted. An atlas that shows
only what it is sure of implies a confidence it has not earned, and the gaps are often where
the interesting work is. Several sit exactly where two fields fail to meet: identifying the
hyssop of the Hebrew Bible needs a botanist and a philologist of Biblical Hebrew, and neither
one settles it alone.
Grouped by the shape of the gap rather than its subject: what sort of thing is unresolved.
existence 5
Recorded as unresolved.
Legendre's Conjecture
Is there really always at least one prime number between n squared and (n plus one) squared, for every positive integer n?
Open Legendre's Conjecture
Why It Is Open No proof or counterexample has ever been found, despite the claim being checked by computer for enormous ranges of n. Results exist for closely related but weaker statements, such as a prime between consecutive cubes for large enough n, but nobody has found a way to close the gap for consecutive squares, which is exactly the gap that makes it one of Landau's four unapproachable problems named in 1912.
What Would Settle It A general proof covering every positive integer n, or a single confirmed counterexample: one gap between consecutive squares, however large, that a careful search shows contains no prime at all.
Expertise Needed Analytic number theory
Question posed in
Legendre's Conjecture (Wikipedia), First paragraph.
Beal Conjecture
Whenever A to the x plus B to the y equals C to the z, with x, y and z all greater than two, do A, B and C really always share a common prime factor?
Open Beal Conjecture
Why It Is Open No complete proof or counterexample has been found since Andrew Beal proposed the conjecture in 1993. Partial results confirm it for many specific combinations of exponents, but a general argument covering every combination, or a single counterexample disproving it, has eluded both professional and amateur attempts despite the million dollar prize Beal has offered.
What Would Settle It A general proof covering every valid combination of A, B, C, x, y and z, or a single confirmed counterexample: one solution where A, B and C share no common prime factor.
Expertise Needed Number theory
Question posed in
Beal Conjecture (Wikipedia), Statement section.
Erdos-Straus Conjecture
Does the Erdos-Straus equation 4/n = 1/x + 1/y + 1/z really have a positive-integer solution for every integer n of 2 or more?
Open Erdos-Straus Conjecture
Why It Is Open Computer search has confirmed the equation solvable for every n up to 10^17, and modular identities cover infinitely many residue classes, but no argument closes the remaining, conjecturally empty, set of exceptions for every n at once.
What Would Settle It A general proof, or a genuine counterexample, covering every integer n rather than a further extension of the verified range.
Expertise Needed Number theory
Question posed in
Erdos-Straus Conjecture (Wikipedia), Verification and progress sections.
Lonely Runner Conjecture
Does every runner on a circular track, moving at its own constant and distinct speed, really become lonely at some moment, for every number of runners n?
Open Lonely Runner Conjecture
Why It Is Open The conjecture is proved individually for up to 13 runners, most recently n = 10 through 13 in 2025 and 2026, and Tao proved an improved general asymptotic bound in 2018, but no argument covers every n at once.
What Would Settle It A proof, or a counterexample, covering every number of runners n rather than a further specific value.
Expertise Needed Combinatorics
Question posed in
Lonely Runner Conjecture (Wikipedia), For specific n section, and Tighter bounds section.
Union-Closed Sets Conjecture
Does every finite union-closed family of sets, other than the family holding only the empty set, really contain an element belonging to at least half of the sets in the family?
Open Union-Closed Sets Conjecture
Why It Is Open The conjecture is confirmed for families of at most 50 sets and other special cases, and Gilmer's 2022 argument reaches only a weaker constant near 0.38 rather than 0.5, so no proof reaches the full one-half bound for every family.
What Would Settle It A proof that some element always belongs to at least half of the sets in any union-closed family, or a counterexample family where no element does.
Expertise Needed Combinatorics
Question posed in
Union-Closed Sets Conjecture (Wikipedia), lead section and Partial results section.
conceptual 1
Recorded as unresolved.
Continuum Hypothesis
Is there really a size of infinity strictly between the integers and the real numbers, and if not, does that fact hold absolutely or only relative to which further axioms mathematicians choose to accept?
Open Continuum Hypothesis
Why It Is Open Godel (1940) and Cohen (1963) proved that the continuum hypothesis can be neither proved nor disproved from the standard ZFC axioms of set theory. That result closes the question of what ZFC alone can say, but it opens a harder one: whether CH has a determinate truth value at all, and if so what further axiom would reveal it. Mathematicians who accept that set theory describes a single true universe of sets, Godel among them, hold this is a real unanswered question rather than a dead end; committed formalists tend to hold there is nothing further to ask.
What Would Settle It Either a broadly accepted new axiom for set theory, beyond ZFC, that settles CH one way or the other and gains the kind of consensus ZFC itself enjoys, or a philosophical argument persuasive enough to convince most set theorists that the independence result is the end of the matter rather than the start of a harder question.
Expertise Needed Mathematical logic and set theory
Question posed in
Continuum Hypothesis (Wikipedia), Arguments for and against section.
Not Yet Classified 10
Recorded as unresolved.
Riemann Hypothesis
Do all nontrivial zeros of the Riemann zeta function really lie on the critical line?
Open Riemann Hypothesis
Why It Is Open No proof or disproof has been found since Riemann first conjectured it in 1859, despite it being one of the most heavily attacked problems in mathematics; computation has verified the first many trillions of zeros with no exception, which is strong evidence but not a proof.
What Would Settle It A general proof (or a single confirmed counterexample, a nontrivial zero found off the critical line) covering every zero, not merely the ones checked so far; a Clay Mathematics Institute Millennium Prize of one million dollars is offered for a correct resolution either way.
Expertise Needed Analytic number theory
Question posed in
Clay Mathematics Institute.
P versus NP
Is every efficiently checkable problem also efficiently solvable, or is checking genuinely easier than solving?
Open P versus NP
Why It Is Open Fifty years of concerted effort by theoretical computer scientists has produced neither a proof that P equals NP nor a proof that it does not; most researchers believe P does not equal NP but this remains an unproven belief, not a result.
What Would Settle It A proof either that some NP problem provably cannot be solved in polynomial time (P does not equal NP), or a genuine polynomial-time algorithm for an NP-complete problem (P equals NP); a Clay Mathematics Institute Millennium Prize of one million dollars is offered for a correct resolution either way.
Expertise Needed Computational complexity theory
Question posed in
Clay Mathematics Institute.
Goldbach Conjecture
Can every even integer greater than two really always be written as the sum of two primes?
Open Goldbach Conjecture
Why It Is Open Verified by computer for every even number checked so far, into the many quintillions, with no exception found, but no general proof covering all even numbers exists, nearly three centuries after Goldbach first proposed it.
What Would Settle It A general proof covering every even integer (or a single confirmed counterexample, an even number that is not the sum of two primes).
Expertise Needed Analytic number theory
Question posed in
MacTutor History of Mathematics Archive.
Twin Prime Conjecture
Do infinitely many twin prime pairs really exist, or does the last one eventually appear?
Open Twin Prime Conjecture
Why It Is Open 2013's breakthrough (Yitang Zhang, later sharpened by the Polymath collaborative project to a gap of 246) proved infinitely many prime pairs exist within SOME bounded gap, but closing that gap all the way down to exactly two, the original conjecture, remains open.
What Would Settle It A proof narrowing the proven bounded gap (currently 246) all the way to 2, or a fundamentally different argument establishing the gap-2 case directly.
Expertise Needed Analytic number theory
Question posed in
MacTutor History of Mathematics Archive.
Collatz Conjecture
Does the Collatz process really always reach one, from every possible starting number?
Open Collatz Conjecture
Why It Is Open Every starting number ever tested by computer, into the astronomically large, reaches one, but no proof rules out either an unbounded starting number whose path never terminates or a hidden cycle other than the trivial one at 1-4-2; Paul Erdos himself is widely quoted as doubting mathematics currently has the tools to settle it.
What Would Settle It A general proof covering every positive starting integer, or the discovery of a starting number that provably never reaches one (an unbounded trajectory or a second cycle).
Expertise Needed Number theory / dynamical systems
Question posed in
Wolfram MathWorld.
Birch and Swinnerton-Dyer Conjecture
Does the rank of an elliptic curve's group of rational points really equal the order of vanishing of its L-function at s equals one, for every elliptic curve?
Open Birch and Swinnerton-Dyer Conjecture
Why It Is Open Proved only in the rank zero and rank one cases, through work by Coates and Wiles in 1977, Gross and Zagier in 1986, and Kolyvagin in 1989. No general argument is known for higher rank, and no counterexample has ever been found either.
What Would Settle It A general proof covering every rank, or a single explicit elliptic curve whose rank and L-function order of vanishing disagree, reaching beyond the rank zero and rank one cases current methods can prove.
Expertise Needed Number theory
Question posed in
Clay Mathematics Institute, https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf.
Hodge Conjecture
Is every Hodge class on a smooth complex projective algebraic variety really a rational combination of the cohomology classes of actual algebraic subvarieties?
Open Hodge Conjecture
Why It Is Open Proved only in codimension one, by the Lefschetz theorem on (1,1) classes. Higher codimension cases, including named examples such as the diagonal cycle's Kunneth components, remain open, and the natural integral version of the conjecture is known to be false.
What Would Settle It A general proof for every codimension, or an explicit Hodge class shown not to be a combination of algebraic cycle classes.
Expertise Needed Algebraic geometry
Question posed in
Clay Mathematics Institute, https://www.claymath.org/wp-content/uploads/2022/06/hodge.pdf.
Navier-Stokes Existence and Smoothness
Do smooth, globally defined solutions to the three dimensional Navier-Stokes equations always exist, or can a solution break down in finite time?
Open Navier-Stokes Existence and Smoothness
Why It Is Open As the Clay Mathematics Institute's own official problem statement puts it, standard methods from partial differential equations appear inadequate, and whether solutions even exist in general was, at the time of writing, unknown.
What Would Settle It A proof that smooth solutions always exist on ordinary three dimensional space or the three dimensional torus given smooth initial data, or a single explicit case where a solution breaks down in finite time.
Expertise Needed Partial differential equations and fluid dynamics
Question posed in
Clay Mathematics Institute, https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf.
Yang-Mills Existence and Mass Gap
Does a nontrivial quantum Yang-Mills theory really exist for every compact simple gauge group in four dimensions, with a strictly positive mass gap?
Open Yang-Mills Existence and Mass Gap
Why It Is Open The mass gap is well established by physics experiment and by computer simulation, but no one has derived it as a mathematical consequence of the quantum Yang-Mills equations themselves, and no fully rigorous construction of a four dimensional quantum Yang-Mills theory yet exists.
What Would Settle It A rigorous mathematical construction of quantum Yang-Mills theory on four dimensional space for a compact simple gauge group, together with a proof that it has a mass gap greater than zero.
Expertise Needed Mathematical physics
Question posed in
Clay Mathematics Institute, https://www.claymath.org/millennium/yang-mills-the-maths-gap/.
ABC Conjecture
Is the abc conjecture actually true, and does Shinichi Mochizuki's inter-universal Teichmuller theory really prove it?
Open ABC Conjecture
Why It Is Open Mochizuki claims a proof through his inter-universal Teichmuller theory, formally published in 2021 by the journal he edits, but Peter Scholze and Jakob Stix identified what they describe as a serious, unfixable gap around a step called Corollary 3.12 in 2018, and most number theorists have not accepted the proof as correct.
What Would Settle It Either a version of the argument that resolves the Scholze-Stix objection in a way the wider number theory community accepts, or an independent proof, or a counterexample.
Expertise Needed Number theory
Question posed in
Klarreich, Titans of Mathematics Clash Over ABC Conjecture (2018), Erica Klarreich, https://www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/.