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Sendov’s conjecture
Every zero of a degree-at-least-two complex polynomial with all zeros in the closed unit disk should have a nearby critical point, at distance at most one.
Open workspace Active AI research · not a proofOpen conjectureCaccetta–Häggkvist Conjecture
Must every finite loopless simple digraph with minimum outdegree d contain a directed cycle of length at most the ceiling of n divided by d?
Open workspace Active AI research · not a proofRecent proof claim under reviewErdős–Rado Sunflower Conjecture
For three petals, must every sufficiently large uniform set family contain three sets with the same pairwise intersection?
Open workspace Active AI research · not a proofOpen conjectureBarnette's Conjecture
Does every finite simple 3-connected planar graph that is both cubic and bipartite contain a Hamiltonian cycle?
Open workspace Active AI research · not a proofOpen conjectureConway’s Thrackle Conjecture
In a drawing where every pair of distinct edges meets exactly once, the conjecture says the graph cannot have more edges than vertices.
Open workspace Active AI research · not a proofOpen conjectureFröberg’s Conjecture
Does the Hilbert series of an ideal generated by generic homogeneous forms always equal the positive truncation of the expected product formula?
Open workspace Active AI research · not a proofOpen conjectureManickam–Miklós–Singhi Conjecture
If n≥4k real numbers have nonnegative total, must at least a k/n share of all k-subsets also have nonnegative sum?
Open workspace Active AI research · not a proofPartially resolvedAlon–Jaeger–Tarsi Conjecture
Given an invertible matrix over a field with at least four elements, can one choose a vector whose coordinates and transformed coordinates are all nonzero?
Open workspace Active AI research · not a proofOpen conjectureFan–Raspaud Conjecture
Can every finite bridgeless cubic graph admit three perfect matchings with no edge common to all three?
Open workspace Active AI research · not a proofPartially resolvedBrualdi–Hollingsworth Conjecture
Can every one-factorization of a complete graph on 2m vertices, for m at least 3, be repartitioned into m spanning trees that each use every factor color exactly once?
Open workspace Active AI research · not a proofOpen conjectureChern’s Conjecture for Closed Affine Manifolds
Must every closed manifold carrying an affine structure have Euler characteristic zero?
Open workspace Active AI research · not a proofOpen conjectureAgoh–Giuga conjecture
An integer greater than one should satisfy the stated power-sum congruence exactly when it is prime.
Open workspace Active AI research · not a proofOpen conjectureDürer’s Edge-Unfolding Conjecture
Does every convex polyhedron have a spanning tree of its actual edges that lets the surface unfold into the plane without overlapping face interiors?
Open workspace Active AI research · not a proofOpen conjectureAlbertson Conjecture
Must every graph have at least as many crossings as the complete graph whose number of vertices equals the graph's chromatic number?
Open workspace Active AI research · not a proofOpen conjectureChowla’s Cosine Conjecture
For every finite set of positive integer frequencies, must the corresponding cosine sum have a negative value whose magnitude is at least a constant times the square root of the set size?
Open workspace Active AI research · not a proofOpen conjectureConway’s 99-Graph Conjecture
Does a strongly regular graph with 99 vertices, degree 14, one common neighbor for adjacent pairs, and two common neighbors for nonadjacent pairs exist?
Open workspace Active AI research · not a proofOpen conjectureGallai’s Path-Decomposition Conjecture
Can the edges of every finite connected simple graph be partitioned into at most half as many simple paths as vertices, rounded up?
Open workspace Active AI research · not a proofLiterature: finite check remainsGraham’s Rearrangement Conjecture
Can every finite set of nonzero residues modulo a prime be ordered so that all of its nonempty partial sums are different?
Open workspace Active AI research · not a proofOpen conjectureHall’s Random-Triangle Conjecture
Among planar convex regions of a fixed area, does the disk give the largest chance that three independent uniform points form an acute triangle?
Open workspace Active AI research · not a proofOpen conjectureKahane’s Quantitative Beurling–Helson Conjecture
Must a continuous circle phase be affine whenever the Wiener norms of all its large integer powers grow more slowly than logarithmically?
Open workspace Active AI research · not a proofPartially resolvedLane–Emden Conjecture
The conjecture says the coupled Lane–Emden system has no everywhere-positive classical solution on all of Euclidean space when the exponents lie strictly below the critical hyperbola.
Open workspace Active AI research · not a proofOpen conjectureNegami’s Planar Cover Conjecture
Negami’s conjecture says a connected graph has a finite planar cover exactly when it can be embedded in the projective plane.
Open workspace Active AI research · not a proofOpen conjectureOda’s Strong Factorization Conjecture
Given two smooth rational fans in one lattice with the same support, the conjecture asks whether both can be refined to one smooth fan using only ordinary smooth star subdivisions.
Open workspace Active AI research · not a proofOpen conjectureOre-Type Bondy Longest-Cycle Conjecture
In a k-connected graph with the required degree-sum bound, must every component outside a longest cycle avoid paths on k vertices?
Open workspace Active AI research · not a proofOpen conjectureRational Homological Quillen Conjecture at p = 2
For a finite group with no nontrivial normal 2-subgroup, the conjecture predicts nonzero rational reduced homology in the poset of its nontrivial elementary abelian 2-subgroups.
Open workspace Active AI research · not a proofOpen conjectureStahl’s Multichromatic Kneser-Graph Conjecture
When every k-subset of [n] receives s colors and disjoint subsets receive disjoint palettes, Stahl’s conjecture gives the exact minimum total number of colors.
Open workspace Active AI research · not a proofOpen problemRiemann Hypothesis
Do every nontrivial zero of the completed zeta function lie on the critical line with real part one half?
Open workspace Active AI research · not a proofOpen conjectureBateman–Horn Conjecture
How often does every polynomial in a fixed admissible family take a prime value at the same integer input?
Open workspace Active AI research · not a proofRecent proof claim under reviewCramér Prime-Gap Conjecture
Can every sufficiently large gap between consecutive primes be bounded by a constant times the square of the logarithm of the earlier prime?
Open workspace Active AI research · not a proofPartially resolvedFontaine–Mazur Conjecture
Which irreducible p-adic Galois representations satisfying arithmetic finiteness conditions actually come from algebraic geometry?
Open workspace Active AI research · not a proofOpen conjectureRyser's Conjecture for Multipartite Hypergraphs
Can every finite r-partite r-uniform hypergraph be covered by at most r−1 times as many vertices as the number of edges in a largest matching?
Open workspace Active AI research · not a proofRecent proof claim under reviewResolution of Singularities in Positive Characteristic
Over a perfect field k of characteristic p>0, must every integral finite-type k-variety X of dimension at least four admit a proper birational morphism π:Y→X with Y regular?
Open workspace Active AI research · not a proofRecent proof claim under reviewUnique Games Conjecture
Can a computer efficiently distinguish a permutation-constraint graph in which almost all edges can be satisfied from one in which almost none can?
Open workspace Active AI research · not a proofOpen conjectureBaum–Connes Conjecture Without Coefficients
Does the proper geometry of a group recover every K-theory class of its reduced group C*-algebra, and only those classes?
Open workspace Active AI research · not a proofOpen problemBloch–Beilinson Conjectures
Do rational Chow groups carry a canonical finite filtration whose layers reflect topology, motives, and regulators?
Open workspace Active AI research · not a proofOpen conjectureCannon Conjecture
Does every word-hyperbolic group with a two-sphere boundary come from a geometric symmetry group of hyperbolic three-space?
Open workspace Active AI research · not a proofOpen problemExistence of One-Way Functions
Can a deterministic function be efficient to evaluate while every efficient randomized algorithm fails to invert a typical output?
Open workspace Active AI research · not a proofRecent proof claim under reviewGraph Reconstruction Conjecture
Can a finite simple graph on at least three vertices always be recovered, up to isomorphism, from the multiset of all graphs obtained by deleting one vertex?
Open workspace Active AI research · not a proofRecent proof claim under reviewMahler Volume-Product Conjecture
Which convex bodies minimize the affine-invariant product of their volume and the volume of their polar dual?
Open workspace Active AI research · not a proofOpen conjectureQuantum PCP Conjecture
Does approximating the ground energy of a quantum many-body system remain QMA-hard when both interaction locality and the YES/NO energy gap are fixed constants?
Open workspace Active AI research · not a proofOpen problemThree-Dimensional Euler Regularity and Blow-Up
Can every smooth finite-energy three-dimensional incompressible Euler flow remain regular for all time, or can vorticity become singular in finite time?
Open workspace Active AI research · not a proofPartially resolvedAbundance Conjecture
If the canonical divisor of a mildly singular projective variety is already nonnegative on every curve, the conjecture asks whether some multiple has enough global sections to define a morphism with no base points. The source narrows possible counterexamples but reports no full proof.
Open workspace Active AI research · not a proofOpen conjectureArtin’s Primitive-Root Conjecture
Choose an integer that is neither a square nor one of the excluded trivial cases. Artin’s conjecture predicts that it generates every nonzero residue modulo a positive proportion of primes, with a precise density. The source controls some ranges and isolates a tensor-sum bottleneck, but reports no proof of the conjecture.
Open workspace Active AI research · not a proofPartially resolvedBirch and Swinnerton-Dyer Conjecture
Does the behavior of an elliptic curve’s L-function at its central point determine the curve’s rational rank and exact arithmetic invariants?
Open workspace Active AI research · not a proofPartially resolvedBorel Rigidity Conjecture
Is every homotopy equivalence between closed aspherical topological manifolds of dimension at least five deformable to a homeomorphism?
Open workspace Active AI research · not a proofOpen problemExponential Time Hypothesis and Strong ETH
Must satisfiability require exponential time in the worst case, and does the best possible base for fixed-width SAT approach two as the clause width grows?
Open workspace Active AI research · not a proofOpen conjectureFalconer Distance Conjecture
Must a compact set in Euclidean space whose Hausdorff dimension is greater than half the ambient dimension determine a positive-length set of distances?
Open workspace Active AI research · not a proofPartially resolvedFarrell–Jones Conjecture
Do virtually cyclic subgroups contain enough information to reconstruct the algebraic K- and L-theory of every group through the Farrell–Jones assembly map?
Open workspace Active AI research · not a proofOpen conjectureGrothendieck Period Conjecture
Are all algebraic relations among the periods of a motive exactly the relations forced by its motivic structure?
Open workspace Active AI research · not a proofPartially resolvedGrothendieck’s Section Conjecture
Does every continuous splitting of a hyperbolic curve’s étale fundamental-group sequence come from one of the curve’s rational points?
Open workspace Active AI research · not a proofOpen problemGrothendieck’s Standard Conjectures on Algebraic Cycles
Do algebraic cycles obey the same Lefschetz decomposition, projector, equivalence, and positivity structures that cohomology predicts?
Open workspace Active AI research · not a proofOpen conjectureHodge Conjecture
Does every rational cohomology class of Hodge type (p,p) on a smooth projective complex variety come from an algebraic cycle?
Open workspace Active AI research · not a proofOpen problemInverse Galois Problem over ℚ
Does every finite abstract symmetry group occur as the Galois group of some finite Galois extension of the rational numbers?
Open workspace Active AI research · not a proofOpen conjectureKannan–Lovász–Simonovits Conjecture
Does every isotropic log-concave probability distribution have a dimension-independent lower bound on how much boundary is needed to cut off a given share of its mass?
Open workspace Active AI research · not a proofOpen conjectureKaplansky Zero-Divisor Conjecture
Can two nonzero finite linear combinations of elements of a torsion-free group ever multiply to zero in the group algebra over a field?
Open workspace Active AI research · not a proofRecent proof claim under reviewMontgomery Pair-Correlation Conjecture
Do the normalized spacings between high Riemann-zeta zeros follow the sine-kernel pair law predicted by random matrix theory?
Open workspace Active AI research · not a proofOpen conjectureMumford–Tate Conjecture
For an abelian variety over a number field, does the symmetry group seen by every ℓ-adic Galois representation exactly match the symmetry group determined by its rational Hodge tensors?
Open workspace Active AI research · not a proofOpen problemNavier–Stokes Existence and Smoothness
For smooth, rapidly decaying, divergence-free initial flow in three dimensions, must the incompressible Navier–Stokes equations produce a smooth solution for all future time?
Open workspace Active AI research · not a proofPartially resolvedNovikov Conjecture
Are the higher signatures of an oriented manifold invariant under oriented homotopy equivalence for every discrete fundamental group?
Open workspace Active AI research · not a proofOpen conjectureP = BPP Derandomization Conjecture
Can every decision problem that admits an efficient randomized algorithm with bounded two-sided error also be solved efficiently by a deterministic algorithm?
Open workspace Active AI research · not a proofRecent proof claim under reviewP versus NP
Are all decision problems whose proposed solutions can be checked efficiently also solvable efficiently?
Open workspace Active AI research · not a proofOpen conjectureCorrected and Refined Malle Conjecture
How many number-field extensions with a prescribed finite permutation group have discriminant at most a given bound once cyclotomic accumulation and Brauer–Manin obstructions are separated correctly?
Open workspace Active AI research · not a proofOpen conjectureSarnak’s Möbius Disjointness Conjecture
The conjecture says that the Möbius function has asymptotically zero correlation with every observable generated by a topological dynamical system of zero entropy.
Open workspace Active AI research · not a proofOpen conjectureSmooth Four-Dimensional Poincaré Conjecture
Must every smooth closed four-manifold with the homotopy type of the four-sphere actually be smoothly equivalent to the standard four-sphere?
Open workspace Active AI research · not a proofPartially resolvedTate Conjecture
Over a finite field, do algebraic cycles account for every Frobenius-fixed even cohomology class, with no hidden nonsemisimple behavior at the corresponding eigenvalue?
Open workspace Active AI research · not a proofPartially resolvedTate’s Rational Leading-Term Stark Conjecture at s = 0
The rational Stark conjecture predicts that the leading term of an Artin L-function at zero, divided by the matching determinant of logarithms of S-units, is algebraic in the character field and varies correctly under Galois conjugation.
Open workspace Active AI research · not a proofOpen problemVP versus VNP — Permanent versus Determinant
Can the permanent be proved inherently much harder than the determinant in a way strong enough to separate the algebraic complexity classes VP and VNP?
Open workspace Active AI research · not a proofOpen conjectureWeak Cosmic Censorship Conjecture
When gravity collapses matter or spacetime strongly enough to form a singularity, must that singularity generically be hidden behind a horizon rather than visible from far away? Weak cosmic censorship predicts that distant observers still see a complete future null infinity. The retained research packet explores what a first visible singularity would have to look like, but it does not prove the conjecture.
Open workspace Active AI research · not a proofRecent proof claim under reviewWhitehead Asphericity Conjecture
Suppose a space is built only from vertices, edges, and filled-in faces, and has no hidden two-dimensional sphere. Can a connected part cut out of it somehow contain such a sphere? Whitehead's conjecture says no: every connected subcomplex of an aspherical two-complex should itself be aspherical. No proof or counterexample is known in the retained packet.
Open workspace Active AI research · not a proofOpen problemYang–Mills Existence and Mass Gap Problem
Can four-dimensional quantum Yang–Mills theory be constructed rigorously so that its vacuum is isolated from every excited state by a positive amount of energy? The source reports several exact lattice-scale footholds, but it does not claim the required continuum theory or mass gap.
Open workspace Active AI research · not a proofOpen conjectureHilbert–Smith Conjecture
Must every locally compact group that acts faithfully and continuously on a connected finite-dimensional manifold be a Lie group?
Open workspace Active AI research · not a proofOpen conjectureAndrews–Curtis Conjecture
Can every balanced presentation of the trivial group be simplified to the obvious free-basis presentation using four elementary relator moves? The source develops a sharply bounded rank-two frontier, but the unrestricted conjecture remains open.
Open workspace Active AI research · not a proofPartially resolvedArtin Holomorphy Conjecture
A nontrivial irreducible Galois representation has an Artin L-function with meromorphic continuation. The conjecture asks whether it can ever have a pole. This work makes the first binary-icosahedral obstruction finite and explicit, then isolates the missing global analytic lift.
Open workspace Active AI research · not a proofRecent proof claim under reviewBerlekamp’s Domineering-Temperature Conjecture
Lech Mazur's manuscript gives a rectangle-reachable 28-cell Domineering position with exact game {17/8 | -2+*} and temperature 33/16 > 2. The attached Lean development kernel-checks the board–target equality, explicit target thermograph, 30-move replay, and resulting existential statement through its narrow HasValueTemperature interface; unrelated external replication and specialist review remain open.
Open workspace Active AI research · not a proofPartially resolvedCherlin–Zilber Algebraicity Conjecture
Must every infinite simple group of finite Morley rank be an algebraic group over an algebraically closed field?
Open workspace Active AI research · not a proofOpen conjectureDe Giorgi Conjecture in Dimensions 5–8
A bounded solution of the Allen–Cahn equation rises strictly in one direction. Must all of its level surfaces be parallel hyperplanes, making the solution a translated and rotated one-dimensional tanh front? The question remains open here in dimensions 5 through 8.
Open workspace Active AI research · not a proofOpen conjectureDynamic Optimality Conjecture for Splay Trees
Does the simple splay operation serve every binary-search-tree access sequence within a constant factor of the best offline rearrangement? A source-reported accounting reduction sharpens the open question but does not prove it.
Open workspace Active AI research · not a proofOpen problemGeneralized Sato–Tate for Generic Picard-Rank-18 K3 Surfaces
For a K3 surface over the rationals whose transcendental cohomology has rank four and no extra Hodge endomorphisms, do normalized Frobenius classes spread out according to Haar measure on SO(4), or on O(4) when the determinant character is nontrivial?
Open workspace Active AI research · not a proofOpen conjectureGraceful Tree Conjecture
Can every finite tree have distinct vertex labels from 0 through its number of edges so that the edge differences are exactly 1 through that number? The source reports a sharp reduction to a boundary-aware gluing problem involving at most four components, but no universal proof.
Open workspace Active AI research · not a proofOpen problemHilbert’s Sixteenth Problem
Which topological arrangements can real algebraic curves have, and how many limit cycles can a bounded-degree planar polynomial vector field possess?
Open workspace Active AI research · not a proofOpen conjectureMorrison–Kawamata Cone Conjecture
For a projective Q-factorial klt Calabi–Yau pair over a characteristic-zero field, does its numerical automorphism group cover the effective nef cone Nef(X) ∩ Eff(X) by translates of one rational-polyhedral chamber, with distinct translates having disjoint interiors?
Open workspace Active AI research · not a proofRecent proof claim under reviewPerfect Cuboid Problem
Can a rectangular box have integer edge lengths, all three face diagonals integral, and its space diagonal integral too? No example or accepted nonexistence proof is known. The retained source reports substantial arithmetic restrictions and a finite frontier for one three-prime case, not a solution of the full problem.
Open workspace Active AI research · not a proofOpen conjectureVaught’s Conjecture
Must a complete theory in a countable language have at most countably many countable models up to isomorphism, or exactly as many as the real numbers? The source develops a conditional tower of prime approximations and isolates a precise branching-versus-coherence obstruction, but it does not solve the conjecture.
Open workspace Active AI research · not a proofOpen conjectureVojta’s Conjecture over Number Fields
Can proximity to a normal-crossings divisor plus canonical height be controlled by one field discriminant and an arbitrarily small big-height allowance for every algebraic point of bounded degree outside a proper exceptional set?
Open workspace Active AI research · not a proofOpen problemZariski Cancellation for Affine Three-Space in Characteristic Zero
Over an algebraically closed field of characteristic zero, if a finitely generated three-dimensional affine domain becomes four-dimensional affine space after adjoining one variable, was it already affine three-space?
Open workspace Active AI research · not a proofOpen problemGeneralized Sato–Tate for Cubic GL₂-Type Abelian Threefolds
For a generic non-CM modular abelian threefold whose coefficient field has three real embeddings, do the three normalized Frobenius angles vary jointly and independently according to the product Sato–Tate measure?
Open workspace Active AI research · not a proofOpen conjectureKashaev–Murakami–Murakami Volume Conjecture
Does the exponential growth rate of a knot's colored Jones polynomial at roots of unity recover the hyperbolic or simplicial volume of the knot complement?
Open workspace Active AI research · not a proofOpen conjectureGoldfeld's Conjecture
For a fixed elliptic curve over the rationals, do half of its quadratic twists have analytic rank zero, half rank one, and only density zero have rank at least two?
Open workspace Active AI research · not a proofOpen conjectureErdős–Hajnal Conjecture
If a large graph avoids one fixed induced pattern, must it contain a clique or independent set whose size is a fixed positive power of the number of vertices? The general conjecture remains open.
Open workspace Active AI research · not a proofOpen conjectureTutte’s 5-Flow Conjecture
Can every bridgeless graph carry a conserved, nonzero flow using values smaller than five? A 6-flow is known for every bridgeless graph, but the universal 5-flow conjecture remains open.
Open workspace Active AI research · not a proofOpen problemCondorcet Winning Sets: Does Size Three Always Suffice?
Can three candidates always form a committee that no outsider can defeat by at least half the voters? External strict-majority work proves a size-five guarantee, but exact alignment with the stronger equality-counts convention has not been established; the exact size-three question remains open.
Open workspace Active AI research · not a proofOpen conjectureElliott–Halberstam Conjecture
Are primes evenly distributed among reduced residue classes on average for moduli almost as large as x? The conjecture remains open; this source isolates a difficult upper-transition estimate rather than proving it.
Open workspace Active AI research · not a proofOpen conjectureHopf Sign Conjecture for Nonpositive Curvature
Does nonpositive curvature force a predictable sign for the Euler characteristic of every closed even-dimensional manifold? It does in important special cases, but the general conjecture remains open.
Open workspace Active AI research · not a proofOpen problemHilbert’s 13th Problem — Algebraic Form
Can the roots of the general seventh-degree polynomial be built in stages using algebraic functions of only two variables? The known upper bound is three variables; proving that two never suffice remains open.
Open workspace Active AI research · not a proofOpen conjectureHuneke–Wiegand Conjecture
Must a finitely generated torsion-free module over a one-dimensional Gorenstein local domain be free whenever its tensor product with its dual has no torsion?
Open workspace Active AI research · not a proofOpen conjectureHartshorne Complete-Intersection Conjecture
Must every smooth projective variety whose dimension is more than twice its codimension be cut out by exactly as many hypersurfaces as its codimension?
Open workspace Active AI research · not a proofOpen problemTermination of Flips in Arbitrary Dimension
Can an arbitrary Minimal Model Program perform infinitely many flips, or must every flip sequence eventually stop in every dimension?
Open workspace Active AI research · not a proofRecent proof claim under reviewabc conjecture
How large can the sum c=a+b be compared with the product of the distinct primes dividing abc, apart from an arbitrarily small power loss?
Open workspace Active AI research · not a proofOpen conjectureKneser–Poulsen conjecture
If every pair of centers moves no farther apart, must the union of their balls lose volume and their common intersection gain volume, even when the radii differ?
Open workspace Active AI research · not a proofOpen conjectureMatrix-multiplication exponent conjecture ω=2
Can square matrices be multiplied in essentially quadratic arithmetic time, matching the unavoidable cost of reading and writing n² entries?
Open workspace Active AI research · not a proofOpen conjectureSmooth four-dimensional Schoenflies conjecture
Does every smoothly embedded three-sphere split the standard four-sphere into two standard smooth four-balls?
Open workspace Active AI research · not a proofOpen conjectureBombieri–Lang Conjecture
Does a variety of general type over a number field have rational points confined to a proper algebraic subset? The submitted route reaches a conditional numerical threshold only for a double-octic test surface; it is not a proof of that case or of the general conjecture.
Open workspace Active AI research · not a proofOpen conjectureHalperin–Carlsson Toral Rank Conjecture
An almost-free r-dimensional torus action should force at least 2^r total rational cohomology classes. The packet narrows one rank-four algebraic configuration but does not prove rank four or the conjecture in arbitrary rank.
Open workspace Active AI research · not a proofOpen problemStrong KPZ Universality
Many one-dimensional random growth models are expected to share one profile-valued scaling limit, the KPZ fixed point. TASEP provides an established fixed-point theorem, while the published Quastel–Sarkar claim for the KPZ equation and finite-range asymmetric exclusion is withdrawn on arXiv v7 with an author-reported proof gap; the packet does not establish the requested broad nonintegrable universality theorem.
Open workspace Active AI research · not a proofOpen conjectureSlice–Ribbon Conjecture: R-Link and Trisection Program
Every ribbon knot is smoothly slice; the open question asks whether every smoothly slice knot is ribbon. This packet's word manipulations do not yet produce the required embedded framed slides, and they do not explain how an arbitrary standard four-ball slice disk supplies compatible R-link data.
Open workspace Active AI research · not a proofPartially resolvedMatchings–Jack Conjecture
After writing the Jack parameter as α = 1 + b, do all Jack connection coefficients have nonnegative integer coefficients in b, as predicted by a combinatorial interpretation using matchings or maps?
Open workspace Active AI research · not a proofOpen problemYau’s Nodal-Set Upper Bound
For a Laplace eigenfunction on any smooth compact Riemannian manifold, is the hypersurface measure of its zero set bounded above by a constant times its frequency √λ?
Open workspace Active AI research · not a proofOpen conjectureChowla’s Conjecture for Liouville Correlations
For any distinct fixed shifts, does the average product of the Liouville signs at those shifted integers tend to zero?
Open workspace Active AI research · not a proofOpen problemElliptic-Curve Discrete Logarithm Challenge Instance
For one specified point P of large prime order and one target point Q on a finite-field elliptic curve, recover the unique scalar x with Q = [x]P.
Open workspace Active AI research · not a proofOpen conjectureCampana–Peternell Conjecture
Must a smooth complex Fano manifold whose tangent bundle is nef be a rational homogeneous space? The source reports conditional progress on cotangent fibers and caustics, but several independent global gates remain open.
Open workspace Active AI research · not a proofOpen conjectureReinhardt Conjecture
Among centrally symmetric convex disks, is the smoothed octagon the unique affine shape with the lowest optimal lattice-packing density? The source reports substantial finite-dimensional reductions but no full proof.
Open workspace Active AI research · not a proofOpen problemPolynomial Entire Minimal Graphs
Nonlinear smooth entire minimal graphs exist in high dimensions, but can one be the graph of a polynomial? The source has no example and narrows only conditional leading-term branches.
Open workspace Active AI research · not a proofSolvedVery General Cubic Fourfold Irrationality — Retained Monodromy Models
The packet tries to rule out a rational parametrization by forcing a Hodge-theoretic surface carrier and eliminating possible carrier models. Although it postdates the cited 2025/2026 proof sources, its distinct route remains conditional and is not that external proof.
Open workspace Active AI research · not a proofOpen conjectureGreen–Griffiths–Lang Conjecture
Must every nonconstant entire curve on a smooth complex projective variety of general type lie in one proper algebraic subset?
Open workspace Active AI research · not a proofOpen conjectureLegendre Conjecture
Is there always a prime strictly between two consecutive positive squares?
Open workspace Active AI research · not a proofOpen conjectureGreenberg's Conjecture in Iwasawa Theory
For a totally real field, should the p-primary class groups in its cyclotomic Z_p-tower have vanishing Iwasawa mu and lambda invariants?
Open workspace Active AI research · not a proofOpen conjectureFrey–Mazur Conjecture
For primes greater than 17, does the p-torsion Galois module of an elliptic curve over Q determine its Q-isogeny class?
Open workspace Active AI research · not a proofOpen problemSoliton Resolution for the Focusing Energy-Critical Wave Equation
At late times, should every bounded non-scattering wave split cleanly into free radiation and finitely many coherent solitons, continuously in time? The general nonradial full-time problem remains open.
Open workspace Active AI research · not a proofOpen conjectureLonely Runner Conjecture
Given finitely many distinct constant speeds around a unit circle, must there be a time when every moving runner is far enough from a stationary reference? The general answer remains unknown.
Open workspace Active AI research · not a proofOpen conjecture11/8 Conjecture
Must a smooth closed spin four-manifold have at least eleven eighths as much second homology as the magnitude of its signature? The general inequality is still open.
Open workspace Active AI research · not a proofOpen conjectureWeight–Monodromy Conjecture
When a variety degenerates over a local field, monodromy organizes its cohomology into layers. The conjecture says each layer has exactly the Frobenius weight predicted by its position. The general mixed-characteristic case remains open.
Open workspace Active AI research · not a proofPartially resolvedErdős–Turán Conjecture on Arithmetic Progressions
Must every infinite set of natural numbers with divergent reciprocal sum contain arithmetic progressions of every finite length? The source says yes is conjectured, not proved.
Open workspace Active AI research · not a proofPartially resolvedGrothendieck–Katz p-Curvature Conjecture
Does vanishing p-curvature for almost every prime force an algebraic connection to have finite monodromy? The general implication remains open.
Open workspace Active AI research · not a proofPartially resolvedHigher-Dimensional Weinstein Conjecture
Must every Reeb vector field on a closed contact manifold have a periodic orbit? Dimension three is known externally, but the unrestricted higher-dimensional problem remains open.
Open workspace Active AI research · not a proofOpen problemThree-Dimensional Ising Universality
Can critical three-dimensional Ising systems be proved to converge to one universal non-Gaussian continuum field with universal exponents? The exact target remains open.
Open workspace Active AI research · not a proofOpen conjectureCollatz conjecture
Starting from any positive integer, repeatedly halve even values and send odd values to (3n+1)/2. The question is whether every orbit reaches 1; the source's two-gate route has not proved this.
Open workspace Active AI research · not a proofOpen problemImprove classical semiprime factorization
Given a large number N known to equal two primes p and q, find the factors with a classical algorithm whose full bit complexity is asymptotically better than the general number field sieve for every allowed factor ratio.
Open workspace Active AI research · not a proofOpen conjectureFourier restriction conjecture
The conjecture asks exactly when the Fourier transform of an ambient function can be meaningfully restricted to a curved surface. The source focuses on a paraboloid model and does not claim a proof.
Open workspace Active AI research · not a proofOpen conjectureSerre uniformity conjecture
For a non-CM elliptic curve over the rational numbers, the conjecture says that every prime larger than 37 gives the full possible symmetry group on the curve's ell-torsion points.
Open workspace Active AI research · not a proofPartially resolvedKontsevich’s Homological Mirror Symmetry Conjecture
For a compact mirror pair, homological mirror symmetry predicts that the symplectic Fukaya category and the algebraic category of twisted perfect complexes encode the same mathematics. Important families are known, but no uniform theorem covers the general compact smooth proper Calabi–Yau setting.
Open workspace Active AI research · not a proofOpen conjectureErdős–Straus Conjecture
The conjecture asks whether every fraction 4/n with n at least 2 splits into three positive unit fractions. Huge finite ranges and many residue classes are known, but no argument covers every integer.
Open workspace Active AI research · not a proofOpen problemUnrestricted C^r Closing Lemma for r ≥ 2
A recurrent orbit returns arbitrarily close to itself. The question is whether an arbitrarily small unrestricted C^r change can always close such recurrence into a periodic orbit when r is at least 2. The classical C^1 theorem does not provide the general higher-regularity result.
Open workspace Active AI research · not a proofOpen problemInvariant Subspace Problem for Complex Hilbert Space
The problem asks whether every bounded operator on an infinite-dimensional complex Hilbert space preserves some nonzero proper closed subspace. Many operator classes do, but no proof or counterexample is known in full generality.
Open workspace Active AI research · not a proofOpen conjectureLeopoldt's Conjecture
Do the p-adic logarithms of a full set of independent units always remain independent?
Open workspace Active AI research · not a proofPartially resolvedBochner–Riesz Conjecture
How much smoothing at the boundary of the frequency ball is enough for bounded Fourier reconstruction on L^p?
Open workspace Active AI research · not a proofOpen problemL versus NL
Can every nondeterministic logspace computation be simulated in deterministic logspace?
Open workspace Active AI research · not a proofOpen conjectureSlice–Ribbon Conjecture
Does every knot bounding a smooth disk in the four-ball also bound one with no interior maxima?
Open workspace Active AI research · not a proofOpen problemThe Missing Moore Graph
A Moore graph of degree 57 and diameter 2 would have 3,250 vertices and extremal local structure: adjacent vertices share no common neighbor, while each nonadjacent pair shares exactly one. This last missing parameter case remains open.
Open workspace Active AI research · not a proofOpen problemHilbert’s Tenth Problem over Q
Given any polynomial equation with integer coefficients, can an algorithm always determine whether it has a rational solution? The answer over Q remains unknown. The retained handoff proposes an undecidability reduction but explicitly leaves its decisive normalization lemma open.
Open workspace Active AI research · not a proofOpen conjectureFinitistic Dimension Conjecture
Each module with finite projective dimension has some finite resolution length. The conjecture asks whether, for each finite-dimensional algebra, all those finite lengths share one finite upper bound.
Open workspace Active AI research · not a proofOpen problemP versus NC
P captures efficient sequential computation; NC captures computation that can be organized into polynomially many parallel operations and polylogarithmic depth. Whether the two classes are equal remains open.
Open workspace Active AI research · not a proofOpen problemCondorcet Winning Sets
Can every election be represented by at most three candidates so that no outsider is preferred to all three by a strict majority? The size-three guarantee is open. Current external literature gives a general size-five guarantee.
Open workspace Active AI research · not a proofOpen conjectureBirkhoff–Poritsky billiard conjecture
Does a complete smooth family of invariant curves close to the boundary of a strictly convex billiard force the boundary itself to be an ellipse?
Open workspace Active AI research · not a proofOpen conjectureThomas–Yau conjecture
Should categorical stability select a unique special Lagrangian and govern a long-time Lagrangian mean-curvature flow, while instability yields ordered special-Lagrangian factors?
Open workspace Active AI research · not a proofOpen conjectureHopf positive-curvature conjecture
Must every closed even-dimensional manifold whose every tangent two-plane has positive sectional curvature also have positive Euler characteristic?
Open workspace Active AI research · not a proofOpen problemLanglands functoriality: GL4 × GL2 to GL8
Do the local tensor products of a cuspidal GL4 representation and a cuspidal GL2 representation come from one global automorphic representation on GL8, with matching parameters at every place?
Open workspace Active AI research · not a proofOpen conjectureRota’s Basis Conjecture
Given n disjoint bases of a rank-n matroid, can all n² elements always be rearranged into n new bases, each taking exactly one element from every original basis? The source advances one conditional branch but explicitly leaves the general conjecture open.
Open workspace Active AI research · not a proofOpen conjectureHadamard Conjecture
Does a square plus-or-minus-one matrix with mutually orthogonal rows exist in every positive order divisible by four? The source reports audited computations and reductions for a proposed lattice-switching program, but it explicitly leaves the universal construction open.
Open workspace Active AI research · not a proofOpen conjectureSchanuel’s Conjecture
If z1,…,zn are linearly independent over the rationals, must the numbers z1,…,zn and their exponentials together contain at least n algebraically independent quantities? The source develops conditional reductions and a fixed test case, but explicitly leaves both the test case and the full conjecture open.
Open workspace Active AI research · not a proofOpen problemSmale’s Ninth Problem
Can every linear program be solved using a number of arithmetic and comparison operations polynomial only in the number of constraints and variables, independent of coefficient magnitudes? The source gives exact reductions and special-purpose progress but explicitly leaves the general problem open.
Open workspace Active AI research · not a proofOpen problemHilbert’s Twelfth Problem
Can every finite abelian extension of every number field be generated effectively by explicitly prescribed algebraic values of analytic or automorphic functions, with the Artin reciprocity action visible on those values?
Open workspace Active AI research · not a proofOpen conjectureHigher-Dimensional Symplectic Ball-Packing Conjecture
For symplectic balls in real dimension at least six, are total volume and the pairwise two-ball nonsqueezing inequalities the only obstructions to packing them strictly into a larger ball?
Open workspace Active AI research · not a proofOpen conjectureSinger Conjecture for L²-Betti Numbers
For every closed aspherical n-manifold, must all L²-Betti numbers vanish outside the middle dimension, and hence vanish in every degree when n is odd?
Open workspace Active AI research · not a proofOpen problemBott Rational Ellipticity in Dimension Four
Must every closed simply connected four-manifold with nonnegative sectional curvature have second Betti number at most two, equivalently be rationally elliptic?
Open workspace Active AI research · not a proofOpen conjectureLehmer’s Conjecture on Mahler Measure
Can a nonzero, noncyclotomic integer polynomial have Mahler measure arbitrarily close to one?
Open workspace Active AI research · not a proofPartially resolvedIgusa–Denef–Loeser Monodromy Conjecture
Must every pole predicted by resolution data appear as monodromy somewhere arbitrarily near the singular point?
Open workspace Active AI research · not a proofPartially resolvedGeneralized Sato–Tate Conjecture
Do normalized Frobenius conjugacy classes spread through the correct compact symmetry group according to Haar measure?
Open workspace Active AI research · not a proofOpen conjectureFurstenberg’s ×2, ×3 Conjecture
Can a genuinely diffuse ergodic measure be invariant under both doubling and tripling without being uniform Lebesgue measure?
Open workspace Active AI research · not a proofOpen problemPlanar Self-Avoiding-Walk Scaling Limit
Do long critical self-avoiding lattice paths converge to the conformally invariant SLE₈/₃ random curve?
Open workspace Active AI research · not a proofOpen conjectureManin–Peyre Conjecture on Rational Points
After removing a geometrically defined thin exceptional set, do rational points of bounded anticanonical height on a suitable Fano variety follow the predicted Peyre asymptotic?
Open workspace Active AI research · not a proofOpen conjectureLittlewood Conjecture
For every real pair alpha and beta, must n times the product of their two nearest-integer errors become arbitrarily small?
Open workspace Active AI research · not a proofOpen conjectureBombieri–Dwork Conjecture for G-functions
Does every minimal differential equation of a G-function arise from algebraic geometry through Gauss–Manin or Picard–Fuchs constructions?
Open workspace Active AI research · not a proofRecent proof claim under reviewIrrationality of a Very General Cubic Fourfold
Is a very general smooth degree-three hypersurface in complex projective five-space nonrational?
Open workspace Active AI research · not a proofOpen conjectureBott Conjecture on Rational Ellipticity
Must every closed simply connected manifold admitting nonnegative sectional curvature have only finitely many nonzero rational homotopy groups?
Open workspace Active AI research · not a proofOpen conjectureTotal Coloring Conjecture
Can every finite simple graph have its vertices and edges colored together using at most two more colors than its maximum degree? The packet narrows one internal branch to twelve explicit marked configurations, but it neither colors those configurations nor proves the conjecture in general.
Open workspace Active AI research · not a proofOpen conjectureGoldbach's Conjecture
Every even integer at least four is conjectured to be the sum of two primes. This source turns that question into positivity of an exact weighted count and narrows its preferred analytic route to a difficult multiplier-collar estimate, but it contains no proof.
Open workspace Active AI research · not a proofOpen conjectureLog-Brunn–Minkowski Conjecture
The conjecture says a logarithmic interpolation of two symmetric convex bodies should have at least the geometric-mean volume. This packet turns the question into concavity of box sections and proves some source-contained cap configurations, but missing certificates and an unequal-overlap case block broader conclusions.
Open workspace Active AI research · not a proofOpen conjectureKashaev–Murakami–Murakami Volume Conjecture
The conjecture predicts that quantum knot invariants grow at a rate set by the hyperbolic volume of the knot complement. The packet builds exact odd-level and boundary-state machinery, but still lacks universal asymptotics and an even-level geometric state sum, so the full all-integer limit is open.
Open workspace Active AI research · not a proofOpen conjectureApproval-Core Nonemptiness Conjecture
Must every approval election contain a committee that no nonempty proposal of size at most k can block with its proportional coalition of strict gainers?
Open workspace Active AI research · not a proofOpen conjectureBeal Conjecture
Can three pairwise-coprime positive integers satisfy a sum of perfect powers when all three exponents exceed two?
Open workspace Active AI research · not a proofOpen conjectureBrennan's Conjecture
How strongly can the derivative of a conformal map blow up near a wild boundary while still remaining integrable?
Open workspace Active AI research · not a proofRecent proof claim under reviewCasas–Alvero Conjecture
If a polynomial shares a zero with every one of its derivatives, must all of its zeros coincide?
Open workspace Active AI research · not a proofOpen problemElliptic Curves over ℚ of Rank at Least 30
Can one exhibit a nonsingular elliptic curve over ℚ with Mordell–Weil rank at least 30, together with an exact certificate? The maintained record is rank at least 29, so the target remains open.
Open workspace Active AI research · not a proofOpen conjectureEquivariant Tamagawa Number Conjecture
A canonical equivariant arithmetic class packages leading L-values, periods, regulators, and integral data. ETNC predicts that this class vanishes in the appropriate relative K-group; the general conjecture remains open despite many special cases.
Open workspace Active AI research · not a proofOpen conjectureGeneralized Ramanujan Conjecture
Is every local component of every unitary cuspidal automorphic representation of GL_n over a number field tempered? This is known over function fields and has strong partial bounds and density results over number fields, but the number-field conjecture remains open.
Open workspace Active AI research · not a proofOpen conjectureHadwiger–Boltyanski Illumination Conjecture
Can the boundary of every full-dimensional convex body in ℝ^d be illuminated by at most 2^d directions, with equality only for parallelotopes? The conjecture is known for many special families but remains open in general.
Open workspace Active AI research · not a proofOpen conjectureHadwiger's Conjecture
Must every graph be colorable with no more colors than the size of its largest complete minor? The source reports tightly scoped separator reductions and finite certificate work, but the general conjecture remains open.
Open workspace Active AI research · not a proofOpen problemHeesch's Problem
Can plane figures have arbitrarily many complete surrounding layers without tiling the plane—and, more strongly, can every positive finite layer count occur? The source reports a narrow exact lattice computation, while both the standard problem and its stronger exact-value target remain open.
Open workspace Active AI research · not a proofOpen problemDimension-Four Hirsch: Minimal Corridor Census
A hypothetical minimal dimension-four violation would contain a long dual path of tetrahedral facets. The source reports a small-n census of abstract path skeletons that pass necessary tests, but completing any skeleton to a polytopal sphere—or ruling them all out—remains open.
Open workspace Active AI research · not a proofOpen problemDimension-Four Hirsch: Normalization-Fiber Audit
Normalization can split singular local sheets and turn a labeled pseudomanifold into a genuine sphere with more vertices. The source reports reversing two such fibers to diagnose the original singular objects, but neither object is a dimension-four Hirsch counterexample.
Open workspace Active AI research · not a proofOpen conjectureJones Unknot Conjecture
Does Jones polynomial 1 force a knot to be the unknot? The question remains open. It is verified through 24 crossings, while the retained source studies only a restricted weighted-prism program with explicit global gaps.
Open workspace Active AI research · not a proofSolvedLarge Steiner Systems Construction Problem
The requested S(6,7,19) cannot exist. Any such design would yield an S(4,5,17), and the latter was ruled out by a peer-reviewed exhaustive classification-based search in 2008.
Open workspace Active AI research · not a proofOpen conjectureLog-Rank Conjecture
Can two parties compute an entry of every low-rank sign matrix using only a polynomial in the logarithm of its rank many bits? The conjecture is open; the best current general upper bound is O(sqrt(r)).
Open workspace Active AI research · not a proofOpen conjectureMorton–Silverman Uniform Boundedness Conjecture
For fixed dimension, map degree, and number-field degree, should one bound control every rational preperiodic point of every such dynamical system? The conjecture remains open, including the exact-period obstruction for quadratic polynomials.
Open workspace Active AI research · not a proofOpen conjectureNearby Lagrangian Conjecture
For a closed connected smooth manifold Q, must every closed exact Lagrangian inside T*Q be deformable to the zero section by a Hamiltonian isotopy?
Open workspace Active AI research · not a proofOpen problemNo-three-in-line problem
How many points can be selected from an n by n integer grid without ever placing three selected points on one affine line?
Open workspace Active AI research · not a proofOpen conjectureOdd Perfect Number Conjecture
A perfect number equals the sum of its proper divisors; the conjecture says that no odd integer can have this property.
Open workspace Active AI research · not a proofOpen conjecturePólya's Conjecture for the Dirichlet Laplacian
For every bounded planar drum, should each Dirichlet eigenvalue stay above the area-scaled Weyl prediction?
Open workspace Active AI research · not a proofOpen problemSensitivity versus Degree for Boolean Multilinear Polynomials
Find one Boolean polynomial whose number of sensitive coordinates grows faster relative to its degree than the best explicit construction currently recorded. The packet has sharply reduced one finite search, but no qualifying polynomial is known.
Open workspace Active AI research · not a proofOpen conjectureSeymour’s Second Neighborhood Conjecture
In every oriented graph, must some vertex reach at least as many new vertices in exactly two steps as it reaches in one step? Many special and local cases are known, but the general statement remains open.
Open workspace Active AI research · not a proofOpen conjectureSidorenko’s Conjecture
A random-like graph is conjectured to minimize the density of every fixed bipartite pattern among graphs with the same edge density. Many graph families and kernel classes are known, but the universal inequality is still open.
Open workspace Active AI research · not a proofOpen conjectureTwin Prime Conjecture
Do infinitely many primes occur two apart? Bounded-gap theorems show infinitely many prime pairs within a fixed finite distance, but the exact gap two remains unproved.
Open workspace Active AI research · not a proofOpen conjectureUlam’s Packing Conjecture
Is the Euclidean ball the three-dimensional convex body whose densest congruent packing has the lowest possible density?
Open workspace Active AI research · not a proofOpen conjectureUnion-Closed Sets (Frankl) Conjecture
Must every finite nontrivial union-closed family contain an element that belongs to at least half of its sets?
Open workspace Active AI research · not a proofOpen conjectureZauner’s Conjecture and SIC-POVM Existence
Does every complex dimension at least two admit a Weyl–Heisenberg covariant SIC fiducial, preferably with canonical order-three Zauner symmetry?
Open workspace Active AI research · not a proofOpen problemSquare-Freeness of Fermat Numbers
Can any prime divide a Fermat number twice? The packet narrows where such a square factor could occur, but it does not rule one out.
Open workspace Active AI research · not a proofOpen problemMoser’s Worm Problem
How little area can a convex planar shape have while still fitting every curve of length one after moving it? Current certified lower and upper bounds leave a substantial gap.
Open workspace Active AI research · not a proofOpen conjectureZeeman Conjecture
Does multiplying any finite contractible two-dimensional complex by an interval always make it collapsible? Restricted algebraic and fake-surface routes do not yet bridge back to arbitrary complexes.
Open workspace Active AI research · not a proofOpen problemCancellation-Conditioned Amplitude Boundary
The source reports several controlled projector and contraction cases, but no syntax-independent theorem for arbitrary sealed or nonprojective operations.
Open workspace Active AI research · not a proofOpen conjectureDeterministic k-Server Conjecture
Local configurations are sharply controlled, but the proof still lacks a global invariant for changing charts in arbitrary metrics.
Open workspace Active AI research · not a proofOpen problemDoubly Efficient Private Information Retrieval
Known DEPIR constructions use stronger structure; this source asks for an ordinary-LWE construction and maps the barriers any candidate must avoid.
Open workspace Active AI research · not a proofOpen problemErdős–Ulam Problem
Fresh prime and squareclass patterns alone can occur in collinear models, so the missing ingredient must use genuinely planar direction geometry.
Open workspace Active AI research · not a proofOpen conjectureHarborth's Conjecture
Many local gadgets and graph classes are controlled; the unresolved step is fitting them together while preserving planarity and target neighborhoods.
Open workspace Active AI research · not a proofOpen problemInscribed Square Problem
Modern Floer and persistence tools prove broad rectangle results and important special cases; the arbitrary Jordan-curve square remains open.
Open workspace Active AI research · not a proofOpen problemLoss-to-Time State-Preserving Quantum Extraction
The source rules out the unrestricted scalar-loss compiler, proves several conditional compiler routes, and classifies the current deterministic RPO endpoint as already additive. The leading open work is a real lossy-source theorem, a complete concrete parameter ledger, and stronger composition and public-output interfaces.
Open workspace Active AI research · not a proofOpen conjectureLovász Conjecture
Graph symmetry solves many special cases, yet no theorem currently turns all vertex-transitive graphs into one reusable Hamiltonian construction.
Open workspace Active AI research · not a proofOpen problemMagic Square of Squares
Large exact searches and arithmetic reductions rule out broad families, yet no square or impossibility proof is known.
Open workspace Active AI research · not a proofOpen problemSmale's Mean Value Problem
Exact centered cases and low-support algebra are strong, yet the sharp inequality for arbitrary complex polynomials remains open.
Open workspace Active AI research · not a proofOpen conjectureAanderaa–Karp–Rosenberg Conjecture
Must every nontrivial monotone graph property sometimes inspect every possible edge before deciding whether the property holds?
Open workspace Active AI research · not a proofOpen conjecturePillai's Conjecture
For a fixed nonzero gap, can only finitely many pairs of perfect powers differ by exactly that amount?
Open workspace Active AI research · not a proofOpen problemBorsuk problem in four dimensions
Five vertices of a regular four-dimensional simplex already require five colors. The open question is whether five smaller-diameter pieces always suffice. The source closes several special and canonical face regimes, but its correction leaves noncanonical singleton and edge directions, buffered coverage, receiver selection, and cross-source safety open.
Open workspace Active AI research · not a proofRecent proof claim under reviewBing–Borsuk Conjecture
The Bing–Borsuk conjecture asks whether every finite-dimensional homogeneous metric ANR is a topological manifold. The selected source reports a compact, connected counterexample conditional on an imported Bryant–Ferry existence theorem whose revised construction proof has not been independently line-audited here.
Open workspace Active AI research · not a proofOpen conjectureCereceda's Conjecture
For each fixed degeneracy d and every palette of at least d+2 colors, can any two proper colorings be transformed into one another one vertex at a time using only quadratically many steps?
Open workspace Active AI research · not a proofOpen conjectureErdős–Turán Additive-Basis Conjecture
If a set of nonnegative integers represents every sufficiently large integer as a sum of two of its elements, must some integers have arbitrarily many ordered representations?
Open workspace Active AI research · not a proofOpen problemFive-dimensional kissing number
Forty equal spheres can touch one equal central sphere in five dimensions, while current cited work proves only that no more than forty-four can do so. The exact maximum remains open.
Open workspace Active AI research · not a proofOpen problemGauss Circle Problem
How closely does the number of integer lattice points in a growing disk track the disk's area at the conjectured square-root boundary scale?
Open workspace Active AI research · not a proofRecent proof claim under reviewZaremba’s Conjecture
Is there one universal bound on continued-fraction digits that works for a reduced fraction with every nontrivial denominator?
Open workspace Active AI research · not a proofOpen conjectureFour Exponentials Conjecture
If two complex x-values are rationally independent and two complex y-values are rationally independent, must at least one of the four exponentials formed from their pairwise products be transcendental?
Open workspace Active AI research · not a proofOpen conjectureGilbert–Pollak Conjecture
For a finite set of points in the plane, compare the shortest network that may add Steiner junctions with the ordinary minimum spanning tree. The conjecture says the first length is always at least √3/2 of the second. This packet reports substantial local reductions and certified regions, but no complete proof.
Open workspace Active AI research · not a proofOpen problemZarankiewicz Problem
How many edges can a bipartite graph have while avoiding one fixed complete bipartite pattern?
Open workspace Active AI research · not a proofOpen conjectureGNRS Conjecture
Do all proper minor-closed graph families have uniformly bounded L1 metric distortion?
Open workspace Active AI research · not a proofOpen conjectureGrimm's Conjecture
Can distinct prime divisors always be assigned to a run of consecutive composite numbers?
Open workspace Active AI research · not a proofOpen problemBrocard's Problem
Are 4!+1, 5!+1, and 7!+1 the only factorials one below a square?
Open workspace Active AI research · not a proofOpen problemModern 3SUM Hypothesis
Do cubic-universe integer 3SUM and reasonable-real 3SUM each resist every truly subquadratic algorithm in their stated machine models?
Open workspace Active AI research · not a proofPartially resolvedAlon–Tarsi Latin-Square Conjecture
Do even and odd Latin squares fail to cancel in every even order?
Open workspace Active AI research · not a proofOpen conjectureGilbreath's Conjecture
Do repeated absolute differences of consecutive primes always begin with one?
Open workspace Active AI research · not a proofPartially resolvedOdd Distinct Covering-System Conjecture
Must every distinct covering system include at least one even modulus?
Open workspace Active AI research · not a proofOpen problemProuhet–Tarry–Escott Problem
Do ideal equal-sums-of-like-powers identities exist at every size, including the missing size 11?
Open workspace Active AI research · not a proofPartially resolvedSausage Conjecture
For the still-open dimensions five through forty-one, does a touching straight-line sausage minimize rounded convex-hull volume among finite packings of congruent balls?
Open workspace Active AI research · not a proofOpen conjectureErdős–Gyárfás Conjecture
Graphs of minimum degree at least three are conjectured to contain a cycle of length 4, 8, 16, or another power of two. The packet narrows the obstruction but does not close the global merger step.
Open workspace Active AI research · not a proofOpen conjecturePrime-power conjecture for finite projective planes
Must every finite projective plane have prime-power order? The packet reports substantial incidence, code, and lattice structure, but no route presently turns that structure into a proof for all orders.
Open workspace Active AI research · not a proofRecent proof claim under reviewYau’s First-Eigenvalue Conjecture
For every closed, connected, embedded minimal hypersurface in the unit sphere, determine whether the first nonzero Laplace–Beltrami eigenvalue equals the hypersurface dimension.
Open workspace Active AI research · not a proofOpen conjectureErdős–Mollin–Walsh Conjecture
Can three consecutive positive integers all have every prime factor repeated? The packet says no proof is known, but turns any hypothetical example into a tightly constrained Pell-sequence repair problem.
Open workspace Active AI research · not a proofOpen problemTarski’s Exponential-Function Problem
The problem asks for an algorithm that always decides whether any first-order statement about the real numbers with exponentiation is true. The packet narrows the missing step to exact comparisons between separately described exponential roots, or to a strong new finiteness theorem; it does not supply either one.
Open workspace Active AI research · not a proofOpen conjectureBarker Sequence Conjecture
A Barker sequence is a row of plus and minus signs whose shifted copies stay almost perfectly uncorrelated. The packet reports strong restrictions on any sequence longer than 13, but it does not report a proof that none exists.
Open workspace Active AI research · not a proofOpen conjectureAlperin Weight Conjecture
The source reports substantial reusable reductions and projective-algebra tools, but no nontrivial quasi-isolated block has yet been completed with all stabilizer, scalar, and intermediate-block requirements, and full prime-and-cover coverage remains open.
Open workspace Active AI research · not a proofOpen conjectureTruly Subcubic Exact APSP Conjecture
Sampling, exact carry bookkeeping, and compact target cloning narrow the problem, but the algorithm still needs sparse-target replacements for dense uniformization and low-doubling stages.
Open workspace Active AI research · not a proofOpen conjectureEuclidean Atiyah–Sutcliffe Conjecture 1
Exact formulas organize insertion and collision limits and settle several special geometries, while four global nonvanishing bridges remain open.
Open workspace Active AI research · not a proofOpen problemFinite Lattice Representation Problem
Every finite lattice would be representable if one meet-irreducible deletion gadget could always be built. A competing route would disprove the statement by ruling out the seven-element W23 lattice in every finite group interval. The packet narrows both routes without finishing either.
Open workspace Active AI research · not a proofOpen problemGeneralized Star-Height Problem
The packet turns one long-standing question about regular expressions into several exact smaller-looking questions about two-state recurrences, return loops, parity, and finite-group accumulation. Those reformulations sharpen the search, but none yet resolves the original problem.
Open workspace Active AI research · not a proofOpen problemMinimum Overlap Problem
Choose plus or minus signs on every edge of a complete graph to make every two-way vertex signing have small total interaction. The open question asks whether the best possible worst interaction, divided by n^(3/2), approaches a limit.
Open workspace Active AI research · not a proofOpen problemOptimal Explicit PRGs for Width-3 Permutation Branching Programs
The packet reduces the generator problem to preserving one structured covariance average. Random and partially explicit constructions meet many surrounding requirements, but no uniformly explicit short-seed construction is yet reported for that final average.
Open workspace Active AI research · not a proofOpen problemSmale’s Seventh Problem
For every number N of points, the problem asks for a fast deterministic way to place N distinct rational points on the unit sphere almost as evenly as the best possible arrangement, with only logarithmic excess energy. The packet develops several exact tools and corrects failed shortcuts, but does not yet provide that algorithm.
Open workspace ProofAtlas research advances · evidence varies by resultResearch advancesAI-developed research advances from ProofAtlas
Seven outcomes already produced through ProofAtlas, with checked results and unverified manuscripts shown at their exact evidence status.
Explore advances ProofAtlas formalizationNumber theoryCollatz Predecessor Lower Bounds at Exponent 0.90
A standalone Lean 4 development proves that, for every fixed positive target not divisible by 3, the accelerated-Collatz predecessor count is eventually at least x^0.90, in both real- and natural-cutoff forms. A stronger target-dependent positive-constant x^0.901 bound supplies the exponent reserve. Lean checks the exact theorem chain; two disclosed native finite checks are independently replayed over 344,373,768 exact inequalities. This strengthens the earlier ProofAtlas 0.88 result but does not prove the Collatz conjecture or provide an explicit eventual cutoff.
Open formalization ProofAtlas formalizationEuclidean geometryButterfly Theorem
In the checked nondegenerate butterfly configuration, the opposite-chord intersections X and Y have the original chord midpoint M as their midpoint.
Open formalization ProofAtlas formalizationIncidence geometrySylvester–Gallai Theorem
A finite set of more than two points in ℝ² that is not all on one line has a pair whose line contains no third point of the set.
Open formalization ProofAtlas formalizationLattice geometryAlgebraic Lemma Toward Pick’s Theorem
For any finite cyclic list of lattice vertices inside a coordinate box, its signed shoelace-area sum equals the associated weighted lattice-point sum.
Open formalization ProofAtlas formalizationProjective geometryBrianchon’s Theorem
For six recorded nonzero tangent lines to a nonsingular real projective conic, the three joins of opposite consecutive-line intersections are concurrent.
Open formalization ProofAtlas formalizationTopologyNo Retraction from the Disk onto Its Boundary Circle
No continuous map from the closed complex unit disk to its boundary circle can fix every boundary point.
Open formalization ProofAtlas formalizationGeometryHeron’s Formula — Coordinate Identity
For any three points in ℝ², the square of twice their signed coordinate area equals four times the Heron radicand of their side lengths.
Open formalization ProofAtlas formalizationGeometryNapoleon’s Theorem — Algebraic Core
Given three complex points and ω² − ω + 1 = 0, the centroids of consistently oriented equilateral constructions on their sides form an equilateral triple.
Open formalization ProofAtlas formalizationGraph theoryBrooks’s Theorem
Every finite connected simple graph that is neither complete nor an odd cycle can be vertex-colored using at most its maximum degree many colors.
Open formalization ProofAtlas formalizationGraph theory · linear algebraKirchhoff’s Matrix-Tree Theorem
For any finite simple graph and chosen root, the determinant of the integer reduced Laplacian equals the number of spanning trees.
Open formalization ProofAtlas formalizationEnumerative combinatoricsHook-Length Formula
For every finite Young diagram, multiplying all hook lengths by the number of standard Young tableaux gives the factorial of the number of cells.
Open formalization ProofAtlas formalizationEnumerative graph theoryCayley’s Formula for Labeled Trees
For every n ≥ 1, the number of labeled unrooted trees on the vertex set Fin n is exactly n^(n − 2).
Open formalization ProofAtlas formalizationOrder theoryDilworth’s Theorem
Every finite poset in which each antichain has at most k elements admits a cover of all elements by k chains.
Open formalization ProofAtlas formalizationGraph theoryKönig’s Edge-Coloring Theorem
Every finite bipartite simple graph admits a proper edge coloring using its maximum degree many colors.
Open formalization ProofAtlas formalizationGraph theoryFriendship Theorem
In a finite graph with at least two vertices, exactly one common neighbor for every distinct pair forces a universal hub; every other vertex has exactly one neighbor other than the hub.
Open formalization ProofAtlas formalizationPartitionsEuler Pentagonal Recurrence
For every positive integer n, p(n) equals the exact finite alternating sum of earlier partition numbers at generalized pentagonal offsets 1, 2, 5, 7, ….
Open formalization ProofAtlas formalizationGraph theoryBondy's Minimum-Degree Longest-Cycle Conjecture
A complete Lean formalization of Bondy's minimum-degree longest-cycle conjecture at the audited source commit: the headline theorem has no theorem-valued premise and its 204-module first-party cone reports only standard classical foundations.
Open formalization ProofAtlas formalizationGraph theory12-Vertex Hamilton Counterexample
Kernel-checked Lean proof that an explicit 12-vertex regular bipartite tournament has no Hamilton decomposition. The statement matches the unrestricted formulation recorded by Granet and Liebenau–Pehova.
Open formalization ProofAtlas formalizationCombinatoricsErdős–Szekeres Monotone Subsequence Theorem
Any injective sequence of r · s + 1 values in a linear order contains either a strictly increasing subsequence of length r + 1 or a strictly decreasing one of length s + 1.
Open formalization ProofAtlas formalizationNumber theory · dynamical systemsTao’s Almost-Bounded Collatz Orbits
For every real-valued threshold function f on ℕ that tends to infinity, the positive starting values N whose standard Collatz orbit minimum is strictly below f(N) have logarithmic density one.
Open formalization ProofAtlas formalizationNumber theory · dynamical systemsNatural-Density Collatz Descent in Logarithmic Time
For thresholds tending to infinity along odd inputs, odd-relative-density-one many odd starts descend within 145 · log N Syracuse steps; for thresholds tending to infinity on all positive inputs, ordinary-natural-density-one many positive starts descend within 436 · log N raw Collatz steps.
Open formalization ProofAtlas formalizationNumber theory · dynamical systemsPower-Saving Bound for Logarithmic-Time Collatz Descent
For every N ≥ 15,552, the proportion of natural numbers n < N that do not fall below their starting value within k ≤ log n accelerated Collatz steps is at most 10,000,000 · N⁻¹ᐟ¹⁰⁰.
Open formalization ProofAtlas formalizationNumber theoryWolstenholme’s Theorem
For every prime p > 3, the sum 1 + 1/2 + ⋯ + 1/(p − 1), with reciprocals interpreted modulo p², is 0 modulo p².
Open formalization ProofAtlas formalizationDiophantine approximationPower-Law Phase Gap for Multiples of log₂ 3
There exists a positive constant c such that every positive integer q satisfies c · q⁻¹³³ᐟ¹⁰ ≤ ‖q log₂ 3‖, the distance to the nearest integer.
Open formalization ProofAtlas formalizationAnalysisFourier L¹/L² Compatibility Bridge
The function-level Fourier transform agrees almost everywhere with its L² representative for ℝ → ℂ functions in both L¹ and L².
Open formalization ProofAtlas formalizationComplex analysisSendov's Conjecture
A complete Lean formalization of Sendov's conjecture: every zero of a nonzero complex polynomial of degree at least two, with all zeros in the closed unit disk, lies within distance one of a zero of the derivative.
Open formalization ProofAtlas formalizationConvex geometryMoser's Convex Worm Mixed-Area Lower Bound
Four Lean-checked pointwise area inequalities cover finite polygonal worms and all unit worms, with direct motions or reflections allowed. Their strict rational corollaries give the lower bound 0.23743658226923856768; the exact optimum remains open.
Open formalization Paper or research publication · not formally verifiedResearch paperAn Exact Counterexample to Berlekamp's Temperature-2 Conjecture for Domineering
Lech Mazur's manuscript gives a rectangle-reachable 28-cell Domineering position with exact game {17/8 | -2+*} and temperature 33/16 > 2. The attached Lean development kernel-checks the board–target equality, explicit target thermograph, 30-move replay, and resulting existential statement through its narrow HasValueTemperature interface; unrelated external replication and specialist review remain open.
Read publication Paper or research publication · complete Lean formalization linkedResearch paperA Formalized Proof of Bondy's Minimum-Degree Longest-Cycle Conjecture
Lech Mazur's August 16 paper proves Bondy's minimum-degree longest-cycle conjecture and identifies the exact Lean 4 declaration checked at audited source commit 9e13b044…. The retained release audit covers the 204-module dependency cone, reports no project mathematical axioms or sorry, and records only standard classical foundations; independent acceptance and specialist review remain open.
Read publication Paper or research publication · not formally verifiedResearch paperA Simplified Proof Candidate for Sendov's Conjecture with Exact Rational Certificates
Lech Mazur's public Revision 14 presents an all-degree computer-assisted proof candidate for Sendov's conjecture, supported by exact rational terminal certificates and accompanied by an explicit boundary between checked computation and written analysis.
Read publication Paper or research publication · not formally verifiedResearch paperA Computer-Assisted Proof Candidate for Sendov's Conjecture
Lech Mazur's public Revision 16 presents a self-contained computer-assisted proof candidate for Sendov's conjecture, with exact rational terminal certificates, deterministic regeneration, mutation tests, and an explicit boundary between machine-checked computation and the written analytic reduction.
Read publication Existing theorem in MathlibAnalysis and topologyArzelà–Ascoli Theorem
A pointwise equicontinuous bounded-continuous-function family on a compact domain, with values in a common compact set, has compact closure in the uniform topology.
Open Mathlib theorem Existing theorem in MathlibTopologyBaire Category Theorem
A countable set-indexed intersection of open dense subsets remains dense in a Baire space.
Open Mathlib theorem Existing theorem in MathlibAnalysisBanach Fixed-Point Theorem
A contraction on a complete extended metric space has a fixed point reached by iterating from any chosen point with finite first displacement, together with an explicit geometric error bound.
Open Mathlib theorem Existing theorem in MathlibFunctional analysisBanach Open Mapping Theorem
A surjective continuous semilinear map between complete normed spaces over compatible normed fields is an open map.
Open Mathlib theorem Existing theorem in MathlibFunctional analysisBanach–Alaoglu Theorem
Operator-norm closed balls in the continuous dual are compact in the weak-* topology over a proper scalar field.
Open Mathlib theorem Existing theorem in MathlibFunctional analysisBanach–Steinhaus Theorem
Pointwise boundedness of an arbitrary family of continuous semilinear maps from a complete seminormed space forces a uniform operator-norm bound.
Open Mathlib theorem Existing theorem in MathlibNumber theoryBasel Problem
The reciprocal-square series has sum π²/6.
Open Mathlib theorem Existing theorem in MathlibProbabilityBayes' Theorem
The two conditional directions are related through the same measurable intersection mass.
Open Mathlib theorem Existing theorem in MathlibNumber theoryBertrand's Postulate
Every positive natural number n has a prime p in the half-open interval n < p ≤ 2n.
Open Mathlib theorem Existing theorem in MathlibAlgebraBinomial Theorem
A natural power of a sum in a commutative semiring expands as an exact finite binomial-coefficient sum.
Open Mathlib theorem Existing theorem in MathlibCombinatorics and convexityBirkhoff–von Neumann Theorem
Over a linearly ordered field, finite doubly stochastic square matrices are exactly the convex hull of permutation matrices.
Open Mathlib theorem Existing theorem in MathlibAlgebraBurnside's Lemma
For a finite group action with finite fixed-point subtypes and finite orbit quotient, the total number of fixed incidences equals the number of orbits times the group cardinality.
Open Mathlib theorem Existing theorem in MathlibLogic and set theoryCantor's Theorem
No function from a type to its power set is surjective.
Open Mathlib theorem Existing theorem in MathlibGeometryCarathéodory's Convex Hull Theorem
A convex hull is exactly the union of the hulls generated by finite affine-independent subsets.
Open Mathlib theorem Existing theorem in MathlibComplex analysisCauchy Integral Formula
A complex-Banach-space-valued function differentiable on a closed disk is recovered at any interior point from its circle integral against the Cauchy kernel.
Open Mathlib theorem Existing theorem in MathlibCombinatorics and number theoryCauchy–Davenport Theorem
Two nonempty subsets of a prime cyclic group have a sumset of size at least the smaller of p and |S|+|T|−1.
Open Mathlib theorem Existing theorem in MathlibLinear algebra and analysisCauchy–Schwarz Inequality
The norm of an inner product is at most the product of the two vector norms.
Open Mathlib theorem Existing theorem in MathlibAlgebraCauchy's Theorem for Finite Groups
A prime dividing the cardinality of a finite group is the exact order of some group element.
Open Mathlib theorem Existing theorem in MathlibLinear algebraCayley–Hamilton Theorem
An endomorphism of a finite free module over a commutative ring annihilates its own characteristic polynomial.
Open Mathlib theorem Existing theorem in MathlibProbability and analysisCentral Limit Theorem
Centered, unit-second-moment independent identically distributed real random variables have normalized sums converging in distribution to the standard Gaussian law.
Open Mathlib theorem Existing theorem in MathlibAnalysisChange of Variables Formula
A determinant-weighted source integral equals the integral over the injective differentiable image of a measurable set.
Open Mathlib theorem Existing theorem in MathlibProbability and analysisChebyshev Inequality
A real L² function cannot spend too much finite measure far from the integral reference μ[X]: the measure of deviations of size at least c is bounded by the recorded variance expression divided by c².
Open Mathlib theorem Existing theorem in MathlibNumber theory and algebraChinese Remainder Theorem
For coprime natural moduli m and n, residues modulo m*n form a ring equivalent to ordered pairs of residues modulo m and modulo n.
Open Mathlib theorem Existing theorem in MathlibNumber theoryClassification of Pythagorean Triples
All integer solutions of x² + y² = z² are exactly the scaled Euclidean parameter forms, with the legs interchangeable and either sign allowed for z.
Open Mathlib theorem Existing theorem in MathlibFunctional analysisClosed Graph Theorem
A closed graph forces a linear map between complete normed spaces to be continuous.
Open Mathlib theorem Existing theorem in MathlibLogic and foundationsCompactness Theorem for First-Order Logic
A first-order theory has a model if and only if every finite subtheory has a model.
Open Mathlib theorem Existing theorem in MathlibNumber theoryDirichlet's Theorem on Primes in Arithmetic Progressions
Every invertible residue class modulo a positive natural number contains infinitely many primes.
Open Mathlib theorem Existing theorem in MathlibNumber theoryDivergence of the Reciprocal-Prime Series
The reciprocal family indexed by the subtype of natural primes is not summable.
Open Mathlib theorem Existing theorem in MathlibMeasure theory and analysisDominated Convergence Theorem
Almost-everywhere convergence under one integrable pointwise norm bound forces the Bochner integrals to converge to the integral of the limit.
Open Mathlib theorem Existing theorem in MathlibLogic and foundationsDownward Löwenheim–Skolem Theorem
A sufficiently large nonempty structure has an elementary substructure of any prescribed infinite cardinality between the seed-and-language bounds and the ambient size.
Open Mathlib theorem Existing theorem in MathlibAlgebra and number theoryEisenstein Criterion
A primitive positive-degree polynomial over an integral domain is irreducible when a prime ideal contains every lower coefficient but not the leading coefficient, while its square does not contain the constant coefficient.
Open Mathlib theorem Existing theorem in MathlibCombinatorics and number theoryErdős–Ginzburg–Ziv Theorem
Among at least 2n−1 indexed residues modulo n, exactly n indices can be selected whose values sum to zero.
Open Mathlib theorem Existing theorem in MathlibCombinatoricsErdős–Ko–Rado Theorem
A pairwise-intersecting family of r-element subsets of an n-element set, with r at most half of n, has at most binom(n−1,r−1) members.
Open Mathlib theorem Existing theorem in MathlibCombinatoricsEuler’s Odd–Distinct Partition Theorem
For every natural n, partitions of n into odd parts and partitions of n into distinct parts are equinumerous.
Open Mathlib theorem Existing theorem in MathlibNumber TheoryEuler's Totient Theorem
If x and n are coprime natural numbers, then x raised to Euler's totient φ(n) is congruent to 1 modulo n.
Open Mathlib theorem Existing theorem in MathlibMeasure theory and analysisFatou's Lemma
For measurable ℝ≥0∞-valued functions, the lower integral of the pointwise liminf is at most the liminf of the lower integrals.
Open Mathlib theorem Existing theorem in MathlibNumber theoryFermat's Last Theorem for Exponent Four
Nonzero natural numbers cannot satisfy a⁴ + b⁴ = c⁴.
Open Mathlib theorem Existing theorem in MathlibNumber theoryFermat's Little Theorem
If p is prime and n is coprime to p, then n^(p-1) is congruent to 1 modulo p.
Open Mathlib theorem Existing theorem in MathlibNumber theoryFermat's Two-Square Classification
A natural number is a sum of two squares exactly when all of its 3 mod 4 prime exponents are even.
Open Mathlib theorem Existing theorem in MathlibLinear algebra and analysisFinite-Dimensional Spectral Theorem
A self-adjoint operator on a finite-dimensional real or complex inner-product space acts coordinatewise by its eigenvalues after the source-defined isometric diagonalization.
Open Mathlib theorem Existing theorem in MathlibAlgebraFirst Sylow Theorem
If a prime power divides the order of a finite group, the group has a subgroup of exactly that prime-power order.
Open Mathlib theorem Existing theorem in MathlibAnalysisFourier Inversion Formula
If a function and its Fourier transform are integrable, inverse transformation recovers the function at every point where it is continuous.
Open Mathlib theorem Existing theorem in MathlibAnalysisFréchet–Riesz Representation Theorem
A complete real or complex inner-product space is conjugate-linearly and isometrically equivalent to its continuous dual through the inner product.
Open Mathlib theorem Existing theorem in MathlibCategory theoryFreyd–Mitchell Embedding Theorem
Every abelian category admits a full, faithful functor into modules over some ring that preserves finite limits and finite colimits.
Open Mathlib theorem Existing theorem in MathlibMeasure theory and analysisFubini's Theorem
For an integrable real-normed-space-valued function on a product of two s-finite measure spaces, the product-measure Bochner integral equals the iterated integral in the selected order.
Open Mathlib theorem Existing theorem in MathlibAlgebra and complex analysisFundamental Theorem of Algebra
Every complex polynomial of positive degree has a complex root.
Open Mathlib theorem Existing theorem in MathlibNumber theoryFundamental Theorem of Arithmetic
Any finite list of primes whose product is n is a permutation of Mathlib's canonical prime-factor list for n.
Open Mathlib theorem Existing theorem in MathlibAnalysisFundamental Theorem of Calculus
Under differentiability and integrability hypotheses, the oriented interval integral of a derivative equals the endpoint difference.
Open Mathlib theorem Existing theorem in MathlibAlgebraFundamental Theorem of Galois Theory — Finite Case
For a finite-dimensional Galois extension, intermediate fields and subgroups of the Galois group form mutually inverse, inclusion-reversing correspondences.
Open Mathlib theorem Existing theorem in MathlibAnalysisGrönwall's Inequality
A pointwise right-derivative norm inequality produces an explicit Grönwall bound across the interval.
Open Mathlib theorem Existing theorem in MathlibFunctional analysisHahn–Banach Theorem
Every continuous scalar-valued linear functional on a subspace of a real or complex seminormed space extends to the whole space without changing its operator norm.
Open Mathlib theorem Existing theorem in MathlibCombinatoricsHales–Jewett Theorem
For every finite alphabet and finite color set, some finite word dimension makes every coloring contain a monochromatic combinatorial line.
Open Mathlib theorem Existing theorem in MathlibCombinatoricsHall's Marriage Theorem
A family of finite sets has distinct representatives exactly when every finite subfamily has a union at least as large as the subfamily.
Open Mathlib theorem Existing theorem in MathlibGraph theoryHandshaking Lemma
A finite simple graph has an even number of vertices of odd degree.
Open Mathlib theorem Existing theorem in MathlibTopologyHeine–Borel Theorem
In a proper Hausdorff (pseudo-)metric space, compactness of a set is equivalent to being closed and bounded.
Open Mathlib theorem Existing theorem in MathlibTopologyHeine–Cantor Theorem
Compactness upgrades continuity of a map between uniform spaces to uniform continuity.
Open Mathlib theorem Existing theorem in MathlibGeometryHelly’s Theorem
For a sufficiently large finite family of convex sets in dimension d, nonempty intersection of every d+1 indexed members forces a nonempty total intersection.
Open Mathlib theorem Existing theorem in MathlibAlgebraHilbert Basis Theorem
A commutative Noetherian ring remains Noetherian after adjoining one polynomial variable.
Open Mathlib theorem Existing theorem in MathlibAlgebra and geometryHilbert’s Nullstellensatz
For an ideal of polynomials in finitely many variables over k, evaluated at K-valued points with K algebraically closed, the polynomials vanishing on all common zeros form exactly the radical ideal.
Open Mathlib theorem Existing theorem in MathlibMeasure theory and analysisHölder's Inequality
Conjugate power-integral scales bound the lower integral of the pointwise product of two AEMeasurable ℝ≥0∞-valued functions.
Open Mathlib theorem Existing theorem in MathlibComplex analysisHolomorphic Open Mapping Theorem
A map to ℂ analytic near every point of a preconnected set is constant there or maps every ambient-open subset contained in that set to an open subset of ℂ.
Open Mathlib theorem Existing theorem in MathlibCombinatoricsInclusion–Exclusion Principle
A finite union's cardinality is the alternating sum of the cardinalities of all nonempty subfamily intersections.
Open Mathlib theorem Existing theorem in MathlibNumber theoryInfinitely Many Primes
For every natural-number threshold, there is a prime at or above it.
Open Mathlib theorem Existing theorem in MathlibTopology and analysisIntermediate Value Theorem
A continuous function on a closed interval attains every value between its endpoint values in the selected increasing endpoint orientation.
Open Mathlib theorem Existing theorem in MathlibAnalysisInverse Function Theorem
The strict derivative of Mathlib's constructed local inverse at f a is the inverse continuous linear equivalence f'.symm.
Open Mathlib theorem Existing theorem in MathlibNumber TheoryIrrationality of the Square Root of Two
The Real square root of 2 is irrational.
Open Mathlib theorem Existing theorem in MathlibAnalysisJensen's Inequality
Convexity bounds the value at a normalized finite weighted sum by the corresponding weighted sum of the values.
Open Mathlib theorem Existing theorem in MathlibOrder theoryJordan–Hölder Theorem
Composition series with matching endpoints are equivalent after reindexing their adjacent steps.
Open Mathlib theorem Existing theorem in MathlibOrder theoryKnaster–Tarski Theorem
The fixed points of a monotone self-map of a complete lattice carry their own complete-lattice structure.
Open Mathlib theorem Existing theorem in MathlibAnalysisKrein–Milman Theorem
Every compact convex set in a real Hausdorff locally convex topological vector space is the closure of the convex hull of its extreme points.
Open Mathlib theorem Existing theorem in MathlibCombinatoricsKruskal–Katona Theorem
If a colex initial segment 𝒞 of r-subsets has no more members than an r-uniform family 𝒜, then 𝒞 has no larger immediate shadow.
Open Mathlib theorem Existing theorem in MathlibNumber theoryLagrange's Four-Square Theorem
Every natural number is a sum of four squares of natural numbers.
Open Mathlib theorem Existing theorem in MathlibGroup theoryLagrange's Theorem
The natural-number cardinality of a subgroup divides that of its ambient group.
Open Mathlib theorem Existing theorem in MathlibGeometryLaw of Cosines
For any three points in a real inner-product affine space, the squared distance opposite the angle at p₂ is determined by the two adjacent distances and the cosine of that angle.
Open Mathlib theorem Existing theorem in MathlibGeometryLaw of Sines
For any three points in a real inner-product affine space, the sine of the angle at p₂ times the distance p₂—p₃ equals the sine of the angle at p₁ times the distance p₃—p₁.
Open Mathlib theorem Existing theorem in MathlibComplex analysisLiouville's Theorem
A bounded, everywhere complex-differentiable map between complex normed spaces is constant.
Open Mathlib theorem Existing theorem in MathlibLogic and foundationsŁoś's Theorem
An ultraproduct satisfies a first-order sentence if and only if that sentence holds in ultrafilter-many factors.
Open Mathlib theorem Existing theorem in MathlibComplex analysisMaximum Modulus Principle
A complex-differentiable map whose norm attains a maximum inside an open preconnected domain has constant norm on that domain.
Open Mathlib theorem Existing theorem in MathlibCalculus and analysisMean Value Theorem
Some interior tangent to a differentiable real curve is parallel to the secant joining the interval endpoints.
Open Mathlib theorem Existing theorem in MathlibMeasure theory and analysisMinkowski's Lᵖ Inequality
The extended Lᵖ seminorm of a pointwise sum is bounded by the sum of the two extended Lᵖ seminorms when 1 ≤ p.
Open Mathlib theorem Existing theorem in MathlibNumber theoryMöbius Inversion
Divisor summation by the arithmetic zeta function is inverted by Möbius weighting on ordered factor pairs.
Open Mathlib theorem Existing theorem in MathlibMeasure theory and analysisMonotone Convergence Theorem
The nonnegative extended-real lintegral of the pointwise supremum of a measurable monotone sequence equals the supremum of its lintegrals.
Open Mathlib theorem Existing theorem in MathlibComputability and logicMyhill–Nerode Theorem
A language is regular if and only if its left-quotient function has finite range.
Open Mathlib theorem Existing theorem in MathlibNumber theoryOstrowski's Theorem
Every nontrivial real-valued absolute value on ℚ is equivalent to the standard real absolute value or to a p-adic absolute value for one unique prime.
Open Mathlib theorem Existing theorem in MathlibNumber TheoryPell Equation Solvability
For positive d, the Pell equation has a nontrivial integer solution if and only if d is not a square.
Open Mathlib theorem Existing theorem in MathlibAnalysisPicard–Lindelöf Local Existence Theorem
Local Lipschitz and size control give a solution curve through a permitted initial point on the recorded closed interval.
Open Mathlib theorem Existing theorem in MathlibCombinatoricsPigeonhole Principle
A function from a larger finite type to a smaller one has two distinct inputs with the same image.
Open Mathlib theorem Existing theorem in MathlibAnalysisPoisson Summation Formula
Under compact-sup-norm summability of integer translates and summability of integer Fourier samples, the translate sum at each real x equals the corresponding phase-weighted Fourier-sample sum.
Open Mathlib theorem Existing theorem in MathlibAlgebraPrimitive Element Theorem
Every finite-dimensional separable field extension is generated over its base field by a single element.
Open Mathlib theorem Existing theorem in MathlibEuclidean geometryPtolemy’s Theorem
Four cospherical points ordered by two crossing straight-angle hypotheses satisfy Ptolemy's exact distance-product identity.
Open Mathlib theorem Existing theorem in MathlibGeometry and linear algebraPythagorean Theorem
In a real inner-product space, the squared-norm identity for a vector sum holds exactly when the summands are orthogonal.
Open Mathlib theorem Existing theorem in MathlibNumber theoryQuadratic Reciprocity
For distinct odd primes, the two Legendre symbols are related by the quadratic-reciprocity sign.
Open Mathlib theorem Existing theorem in MathlibAnalysisRademacher's Theorem
A globally Lipschitz map between finite-dimensional real normed spaces is Fréchet-differentiable almost everywhere for an additive Haar measure on its domain.
Open Mathlib theorem Existing theorem in MathlibMeasure theory and analysisRadon–Nikodym Theorem
Under the required Lebesgue decomposition, μ is absolutely continuous with respect to ν exactly when weighting ν by rnDeriv μ ν reconstructs μ.
Open Mathlib theorem Existing theorem in MathlibGeometryRadon's Theorem
Every affinely dependent indexed family admits a complementary split whose two image convex hulls have a common point.
Open Mathlib theorem Existing theorem in MathlibLinear algebraRank–Nullity Theorem
For a linear map under Mathlib's rank-nullity hypothesis, the cardinal ranks of the range and kernel add to the rank of the domain.
Open Mathlib theorem Existing theorem in MathlibComputability and logicRice's Theorem
A computably decidable semantic property containing one partial-recursive function contains every partial-recursive function.
Open Mathlib theorem Existing theorem in MathlibCombinatoricsRoth’s Theorem
The maximum size of a three-term-arithmetic-progression-free subset of {0,…,N−1} is little-o of N.
Open Mathlib theorem Existing theorem in MathlibSet theorySchröder–Bernstein Theorem
Injections in both directions between two types imply a bijection between them.
Open Mathlib theorem Existing theorem in MathlibComplex analysisSchwarz Lemma
An origin-fixing complex-differentiable map between equal-radius balls does not increase the norm of an interior point.
Open Mathlib theorem Existing theorem in MathlibProbabilitySecond Borel–Cantelli Lemma
A mutually independent sequence of measurable events with divergent total measure has a limsup of measure one.
Open Mathlib theorem Existing theorem in MathlibCombinatoricsSperner’s Theorem
An antichain of subsets of an n-element set has at most as many members as a middle layer of the Boolean lattice.
Open Mathlib theorem Existing theorem in MathlibAnalysis and topologyStone–Weierstrass Theorem
A point-separating real subalgebra of continuous functions on a compact space is dense in the entire continuous-function algebra.
Open Mathlib theorem Existing theorem in MathlibProbability and analysisStrong Law of Large Numbers
Empirical averages of pairwise independent identically distributed integrable real random variables converge almost everywhere to their common expectation.
Open Mathlib theorem Existing theorem in MathlibAlgebraStructure Theorem for Finitely Generated Abelian Groups
Every finitely generated abelian group decomposes into a finite-rank free part and finitely many cyclic prime-power parts.
Open Mathlib theorem Existing theorem in MathlibAlgebraSylow Conjugacy and Counting Theorems
The finite number of Sylow p-subgroups is congruent to 1 modulo p.
Open Mathlib theorem Existing theorem in MathlibCombinatoricsSzemerédi’s Regularity Lemma
Every finite simple graph admits an ε-uniform equipartition whose number of parts lies between l and an explicit bound depending only on ε and l.
Open Mathlib theorem Existing theorem in MathlibAnalysisTaylor's Theorem with Lagrange Remainder
The error of a finite Taylor polynomial at x is an (n + 1)-st within-derivative at some interior point, multiplied by the standard power-and-factorial factor.
Open Mathlib theorem Existing theorem in MathlibGeometryThales’ Theorem
Given two endpoints of a diameter of a sphere in a real inner-product affine space, the angle they subtend at a point is right exactly when that point lies on the sphere.
Open Mathlib theorem Existing theorem in MathlibTopologyTietze Extension Theorem — Bounded Real-Valued Form
Every bounded continuous real-valued function along a closed embedding into a normal space has a same-norm bounded continuous extension.
Open Mathlib theorem Existing theorem in MathlibAnalysisTonelli's Theorem
An extended-nonnegative product-space integral equals the iterated integral obtained by integrating first against ν and then against μ.
Open Mathlib theorem Existing theorem in MathlibCombinatoricsTurán’s Theorem
For r > 0, a finite graph has the maximum possible number of edges without an (r+1)-clique exactly when it is isomorphic to the balanced complete r-partite Turán graph.
Open Mathlib theorem Existing theorem in MathlibCombinatoricsTutte’s Perfect-Matching Theorem
A finite simple graph has a perfect matching exactly when deleting any vertex set leaves no more odd components than deleted vertices.
Open Mathlib theorem Existing theorem in MathlibTopologyTychonoff's Theorem
An arbitrary product of compact subsets is compact in the product topology.
Open Mathlib theorem Existing theorem in MathlibLogic and set theoryUncountability of the Continuum
The universal set of real numbers is not countable.
Open Mathlib theorem Existing theorem in MathlibComputability and logicUndecidability of the Halting Problem
For every fixed input, Mathlib's predicate that an encoded program is defined on that input is not computable.
Open Mathlib theorem Existing theorem in MathlibTopologyUrysohn's Lemma
Disjoint closed sets in a normal topological space are separated by a continuous real-valued function taking values in [0,1].
Open Mathlib theorem Existing theorem in MathlibLogic and foundationsWell-Ordering Theorem
Every type admits a linear order whose strict relation is well founded.
Open Mathlib theorem Existing theorem in MathlibNumber theoryWilson's Theorem
For n ≠ 1, primality is equivalent to the cast of (n−1)! being −1 in ZMod n.
Open Mathlib theorem Existing theorem in MathlibCategory theoryYoneda Lemma
Natural transformations from the representable presheaf Hom(−,X) to F are equivalent to elements of F(X).
Open Mathlib theorem Existing theorem in MathlibLogic and foundationsZorn’s Lemma
In a preorder where every chain is bounded above, a maximal element exists.
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