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Project Euler Solutions
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Research papers
All 379 papers →Complete-result manuscripts from the Open Math Problems programme, each with the PDF, LaTeX source, a public verification report and a DOI. Unrefereed preprints by Alper Ferudun.
- EP-278 · Complete proofAn Exact Fixed-Parameter Algorithm for Extremal Unions of Residue Classes
- AIM-GEOMETRY-0175 · Complete negative answerComplex Sectional Curvature Blow-Up under Circle Cheeger Collapse
- AIM-TOPOLOGY-0102 · Complete counterexampleA Four-Point Counterexample to Excision for Directed Cubical Homology of Closure Spaces
- AIM-DYNAMICAL_SYSTEMS-0005 · Complete negative answerMaximal Generic Iterated Galois Images Do Not Determine p-Adic Julia Sets
- AIM-COMBINATORICS-0233 · Complete proofPower-Saving Lower Bounds for Additive Bases of Polynomial Sequences
- AIM-DYNAMICAL_SYSTEMS-0095 · Complete proofRamification Portraits of Rigid Lattès Maps
Show all 379 papersShow fewer
- AIM-GEOMETRY-0263 · Complete proofA Compactness Obstruction to Linear Growth Along Null Geodesics
- AIM-COMBINATORICS-0230 · Complete counterexampleLacunary Counterexamples to a Distinct-Summand Freiman Container Problem
- AIM-GEOMETRY-0274 · Complete counterexampleParallel Nilpotent Endomorphisms Without Parallel Null Vectors
- AIM-ALGEBRAIC_NUMBER_THEORY-0109 · Complete counterexampleA Positive-Rank Elliptic Curve with No Dense Prime
- AIM-FUNCTIONAL_ANALYSIS-0027 · Complete proofForcing Absoluteness of Minimal and Maximal C∗-Tensor Products
- AMR-011-0025 · Complete proofA Free-Uniform-Spanning-Forest Proof of Finite-Index Multiplicativity
- AIM-PROBABILITY-0126 · Complete proofA Slice–Riesz–Bochner Characterization of Joint Brown Determinant Functions
- AIM-ANALYSIS-0015 · Complete proofThe Spectrum of the Hilbert Matrix on Power-Weighted ℓ2 Spaces
- AIM-GEOMETRIC_GROUP_THEORY-0027 · Complete proofFree-by-Cyclic Groups with Unboundedly Many BNS Component Orbits
- AIM-GEOMETRY-0195 · Complete proofAngle-Data Reconstruction for Triangulated Polyhedral Surfaces and a Genus Deficit
- AIM-TOPOLOGY-0203 · Complete proofVariable Critical Exponents on a Fixed Free-Deck Regular Cover
- AMR-011-0004 · Complete proofAdjoining a Haar-Generic Matrix to a Parabolic-Free Subgroup of SL₂(Qₚ)
- AIM-DYNAMICAL_SYSTEMS-0011 · Complete proofMaximal Finite-Set Stabilizers in Thompson's Group T
- AIM-PROBABILITY-0111 · Complete negative answerNonuniformity of Free-Gradient Heat Semigroups under Finite Fisher Information
- AIM-ARITHMETIC_GEOMETRY-0067 · Complete affirmative answerA Seventeen-Dimensional Component of the Hilbert Scheme of Eighteen Points on an Integral Curve
- AIM-ARITHMETIC_GEOMETRY-0078 · Complete proofA Generically Nonreduced Component for Hilbert Function (1,4,10,10)
- AIM-ALGEBRAIC_GEOMETRY-0125 · Complete proofSmooth Hypersurfaces Beyond the Chevalley-Warning Range over Prime Fields
- AIM-REPRESENTATION_THEORY-0023 · Complete negative answerQuantum Symmetric Algebras on Multiple Standard Copies: Structure and Categorical Non-Equivalence
- AIM-SEVERAL_COMPLEX_VARIABLES-0010 · Complete proof (stated special case)An Explicit Rational Homotopy from the Faran Map to a Linear Map in Target Dimension Four
- KOU-21.68 · Complete counterexampleA Semi-Abelian Group of Order 768 That Is Not an M-Group
- OWR-12861-021 · Complete negative answerDirac Graphs Without a Spanning Near-Square of an Odd Cycle: A Negative Answer to a Question of Heinig
- KOU-21.76 · Complete affirmative answerExplicit Closed Elementary Nets That Cannot Be Completed, Including Fields of Transcendence Degree One
- OWR-16164-019 · Complete negative answerA Counterexample to a Conjecture of Sportiello on Coloured Permutations in Convex Shapes
- OWR-1782-009 · Complete counterexample (all-n form)Small Counterexamples to the All-n Form of the Exact Codegree Conjecture for Tight Hamiltonian Cycles
- OWR-14299577-018 · Complete proofDifferences of Signed Representatives: Sharp Answers to Two Questions of Bernert and Arala Santos
- AIM-LOGIC-0084 · Complete counterexample (specified presentation)Sharp Computability Bounds for Nonforking Sections
- AIM-LINEAR_ALGEBRA-0012 · Complete theorem (D+a branch)Weighted Root Deletions and Coefficientwise Toeplitz Positivity
- AIM-COMBINATORICS-0177 · Complete counterexample (fixed label order)A Six-Vertex Counterexample to Vertex-Averaged Ordered Reachability
- AIM-REPRESENTATION_THEORY-0102 · Complete obstruction (three explicit families)A Degree-24 Obstruction to Hopf Structures on Symmetric-Group Supercharacter Spaces
- AIM-COMPUTATION-0073 · Complete scoped theorems (closed model)Strict Likelihood Multimodality and Septic Equations for Two-Component Birkhoff Models
- AIM-PROBABILITY-0108 · Complete scoped polynomial theoremsPolynomial Rigidity and a Sharp Degree Bound for Free Quadratic Variation
- AIM-REPRESENTATION_THEORY-0007 · Complete all-rank fixed-vector proofCompact-Unit Types and Fixed Vectors for Symplectic Similitudes
- AIM-PROBABILITY-0027 · Complete finite-system comparison theoremExplicit Stationary Comparisons for Join-the-Shortest-Queue Approximations
- AIM-ANALYSIS-0138 · Complete zero-location theoremsSharp Zero Regions and Hurwitz Criteria for BMV Trace Polynomials
- AIM-ANALYSIS-0089 · Complete symmetric-weighing theoremSharp Induced-Norm Paving for Symmetric Weighing Matrices
- AIM-ANALYSIS-0164 · Complete normalized-covariance criterionA Diagonal-Jet Criterion for Normalized Gaussian Analytic Covariances
- AIM-ALGEBRAIC_NUMBER_THEORY-0111 · Complete low-degree Hilbert-locus theoremLow-Degree Curves and Smooth Hilbert Loci on the Fermat Fourfold
- AIM-COMBINATORICS-0093 · Exact law and five-cycle counterexampleB-Spline Laws and Nonreal Zeros for Weighted-Cycle Internal DLA
- OPG-46575 · Complete negative answerA Negative Answer to Melnikov's Valency-Variety Problem
- OWR-17474-010 · Complete negative answerA Negative Answer to Bäumler's Question on the Number of Spin-Glass Ground States
- OWR-15428-015 · Complete negative answer and characterizationWhen Is the Set of Non-Completable Partial Probability Tensors Convex?
- OWR-4138-001 · Complete proofA Proof of the Nichols–Stolz Conjecture on the Almost-Sure Spectrum of the Bernoulli Displacement Model
- OWR-730-009 · Conditional resolution (ETH)The Exact Complexity of ε-Dense Steiner Tree
- OWR-14298808-012 · Complete proofThe Critical Window for High-Dimensional Limits of Hyperbolic Poisson k-Plane Processes
- OWR-17290-002 · Complete proof (complete setting)Essential Self-Adjointness of the Laplace–Beltrami Operator for a Family of Non-Regular Almost-Riemannian Structures
- OWR-14298367-003 · Complete proofShadowing and Generalized Hyperbolicity for Dissipative Composition Operators without Bounded Distortion
- AIM-OTHER-0001 · Complete rooted-forest linear-space algorithmFull Linear Homomesy Spaces for Rowmotion on Rooted Forests
- AIM-COMPUTATION-0025 · Explicit exponential multimodality boundsExponential Rarity of Multiple Likelihood Modes in Bivariate Seemingly Unrelated Regression
- AIM-COMBINATORICS-0209 · Effective density characterizationA Finite Gap Bound for Local Progression-Free Density
- OPG-37325 · Logarithmic bounds and negative answerLogarithmic Equivalence Covers of Powers of Cycles
- AIM-PROBABILITY-0009 · Exact forest and cycle inference countsBidirectional Inference Counts on Forests and Cycles
- AIM-OTHER-0060 · Finite induced subgraph realizationFinite Bipartite Graphs as Induced Subgraphs of Subfactor Principal Graphs
- OWR-14299088-013 · Complete counterexampleA Negative Answer to a Question of Basu and Perrucci on Two Multi-Affine Polynomials
- OWR-11578-001 · Complete counterexampleA Negative Answer to Harrach's Question on the Potential Map in the Unified Eddy-Current Formulation
- OWR-13498-011 · Complete proofThe Ford-Circle Packing Has Maximum Area: An Answer to a Question of Propp and Kenyon
- AMR-090-0002-0003 · Proof of two Ramassamy conjecturesEuler Numbers Modulo Powers of Two and Arnold's Sequence: Proof of Two Conjectures of Ramassamy
- AMR-096-0015 · Sharp asymptotics and range counterexampleExcursion Lengths in a Uniform Eulerian Circuit of the Complete Graph: Aldous's Conjecture Holds if and only if i = o(n3/2)
- OWR-14299911-029 · Complete corner-peeling counterexampleA Counterexample to a Corner-Peeling Conjecture for Subcomplexes of ℤ3
- OWR-1323-008 · Partial answer (scoped theorems)Hypercyclic Subspaces on ω and a Partial Answer to a Question of Petersson
- OWR-15436-004 · Partial answer (scoped theorems)Multistationarity of the Altan-Bonnet–Germain Kinetic-Proofreading Network: A Partial Answer to a Question of Rendall
- OWR-15428-014 · Partial answer (degree in classified cases)Partial Results on the Degree of the Algebraic Boundary in Rank-One Tensor Completion
- AMR-021-0015 · Complete counterexampleA Reciprocal-Block Counterexample to the Forsgård–Shapiro Coefficient-Parity Bound
- AMR-021-0014 · Complete counterexampleA Positive-Coefficient Counterexample to the Weighted Forsgård–Shapiro Parity Bound
- AMR-021-0016 · Complete criterionA Finite Moment-Cone Criterion for Derivative-Root Configurations
- OWR-13678-010 · Complete answer (exact dimension formula)The Dimension of Trivariate C1 Splines on Generic Bipyramid Cells
- OWR-12697711-015 · Complete answer (all loopless matroids)The Amoeba Dimension of an Arbitrary Loopless Matroid
- AMR-050-0041 · Complete proof (least period N>4)A Telescoping Formula for Evolute Areas in Elliptic Billiards
- OWR-2489-009 · Complete counterexampleCounterexamples to Conforti's Subtree Conjecture for Mixed-Integer Bipartite Covers
- OWR-1452-024 · Negative answer (Q̄ case open)Counterexamples over F̄p to a Generic Equivalence Problem of Kraft and Russell
- OWR-13750332-001 · Complete proof (both conjectures)Star Transforms Without Type 2 Singular Directions and Conflitti's Conjecture on Elementary Symmetric Polynomials
- OWR-1703876-006 · Partial answer (question 1 solved, sharp)A Proof of Paták's kb+1 Conjecture for Constrained Stars, with Improved Bounds for Complete Graphs
- OWR-15208-008 · Complete negative answerThe Conformal Parametrization of Nonnegatively Curved Surfaces Is Not a Homeomorphism at Integer Smoothness
- OWR-14298580-008 · Complete negative answer (computer-assisted)A Negative Answer to the Strong Geography Question of Alfieri and Binns
- OWR-4798-013 · Complete affirmative answerThe Klee–Novik Complexes B(i,d) Triangulate Si × Bd−i−1
- OWR-3389-016 · Complete negative answerThe Harrell–Stubbe Gap Inequality for Dirichlet Eigenvalues Is Never Saturated
- OWR-1195-004 · Complete negative answerKernels of Signature Operators Twisted by Almost Flat Bundles Are Not Homotopy Invariant
- OWR-3385-008 · Complete proofA Proof of the Below–Krummeck–Richter-Gebert Conjecture on Cyclic Determinant Quotients
- OWR-9790358-016 · Partial answer (quadrilaterals complete)Algebraic Relations Among Mean Value Coordinates: A Complete Answer for Quadrilaterals and Partial Results for Larger Polygons
- AMR-046-0027 · Partial answer (existence and small cases)Gasull's Finite Moment Problem for Polynomials with Few Monomials
- OWR-17135-036 · Partial answer (scoped theorems)Combinatorial Spheres inside Spheres on the Same Vertex Set: Partial Results on a Question of Santos
- AMR-014-0019 · Partial answer (scoped theorems)Isolated Real Zeros of Sums of Squares: Quaternary Forms and Quinary Quartics
- AMR-050-0048 · Counterexample and corrected formulaA Focal Inversion Area Ratio for Four-Periodic Elliptic Billiards
- AMR-050-0025 · Complete proofFocal Antipedal Centroids of Rectangles Circumscribed about an Ellipse
- AMR-050-0026 · Complete proofStationary Focal Antipedal Centroids for Even Elliptic Billiard Periods
- OWR-14299906-003 · Negative answer (general case)On a Question of Freslon, Gerontogiannis and Skalski about Quantum Isometries of Cuntz–Krieger Algebras
- OWR-1275-010 · Complete answerComponents and Cohomology of Fock's Baby Teichmüller Space
- OWR-782-001 · Complete proofExtremal Modular Forms Modulo p and a Conjecture of Bannai, Koike, Shinohara and Tagami
- OWR-17294-017 · Negative answer (counterexamples)The Quantum Symmetriser of a Yang–Baxter Solution Need Not Be a Quasi-Isomorphism
- OWR-13497-006 · Complete answerAll Incorrect Equilibria of the Planar Four-Agent Formation Law Are Unstable
- OWR-11786-023 · Complete counterexampleA Negative Answer to a Question of Santos on the Dual Diameter of Surfaces with Boundary
- AMR-050-0068 · Complete scoped proofEqual Focal Inverted Pedal Areas Can Vanish in Elliptic Billiards
- AMR-050-0069 · Complete proofIsogonal Conjugacy and Equal Focal Inverted Pedal Areas
- AMR-050-0001 · Complete counterexamplesFour-Periodic Counterexamples to Two Elliptic Billiard Invariants
- AMR-050-0019 · Complete counterexampleNonconstant Antipedal Area Ratios in Four-Periodic Elliptic Billiards
- AMR-050-0021 · Complete scoped proofFocal Pedal-Antipedal Area Products in Elliptic Billiards
- AMR-050-0012 · Complete scoped proofCenter-Pedal Area Products in Elliptic Billiards
- AMR-050-0014 · Complete scoped proofOuter-Pedal Area Products in Elliptic Billiards
- AMR-050-0027 · Complete scoped proofSteiner-Centroid Pedal Area Ratios in Elliptic Billiards
- AMR-050-0007 · Complete scoped proofFocal Distance Products of Outer Elliptic Billiard Polygons
- AMR-050-0023 · Complete scoped proofOriginal Antipedal Centroids in Elliptic Billiards
- AMR-050-0029 · Complete scoped proofInner Steiner Pedal Ratios and Exceptional Zero-Area Families
- AMR-050-0036 · Complete scoped proofA Six-Periodic Zero of Focal Outer-Antipedal Area
- AMR-050-0044 · Complete scoped proofFocal Inversion Area Products in Elliptic Billiards
- AMR-050-0046 · Complete scoped proofFocal Inversion Area Ratios in Elliptic Billiards
- AMR-050-0062 · Complete proofProducts of the Two Focal Inversion Areas in Elliptic Billiards
- AMR-050-0053 · Complete adjacent-conjecture counterexampleA Pentagonal Counterexample to a Poncelet Squared-Side Criterion
- AIM-GEOMETRIC_GROUP_THEORY-0134 · Exact limit and finite-CW counterexampleExact Heisenberg Determinant Limits and Tower-Dependent Homological Torsion
- OWR-15212-005 · Complete negative answerA Negative Answer to Wolansky's Question on Mutually Dominating Multiphase Measure Spaces
- OWR-12723-008 · Complete answer (on the square)The Sharp Hexagonal Constant in Steinerberger's Geometric Uncertainty Principle on the Square
- OWR-2040-002 · Counterexamples (Conjecture 2 first refuted by Kriebel)Failures of Theorem 1 and Conjecture 3 in Knudson's Persistence–Gradient Abstract, and a Corrected Form of Conjecture 2 after Kriebel
- OWR-16766-004 · Complete counterexampleA Negative Answer to the Kadu–van Leeuwen Conjecture on Dual Binary Tomography
- OWR-15428-003 · Complete negative answerSeigal's Conjecture on Gram Determinants of Real Binary Tensors Fails for n ≥ 4
- AIM-GEOMETRY-0036 · Counterexample and corrected fixed-point proofBalanced Laurent Potentials for Finite Cluster Actions
- AMR-103-0226 · Complete affirmative answerKnots with Signature 4 and Determinant 4k+1: An Answer to a Question of Shinohara
- AIM-GEOMETRY-0250 · Counterexample and sufficient descent criterionMinimal Geodesics under Finite Covers
- AMR-099-0003 · Complete affirmative answerEquilibria of Inverse-Square Repulsion on the Line Are Arithmetic Progressions
- AIM-PROBABILITY-0131 · Bounded-tracial theoremFourier Shadows and Tracial Free Infinite Divisibility
- AIM-ALGEBRAIC_GEOMETRY-0289 · Presentation-independence theoremPresentation Independence of Ambient Adjoint Cartier Algebras
- AMR-022-7062 · Complete negative answerScales of Convergent Series and the Null Ideal: An Answer to Hinkkanen's Problem 7.62
- AMR-022-2045 · Complete affirmative answer (computer-assisted)The Equation J0(z) = 1 Has at Most One Solution on Each Ray: An Answer to a Question of Zalcman
- AMR-066-0018 · Complete counterexampleNonflat Metrics with Nonnegative Scalar Curvature That Are Flat at Infinity: A Counterexample to a Conjecture of Gromov
- AIM-LINEAR_ALGEBRA-0007 · Regular-loop logarithm counterexampleRegular Totally Nonnegative Loops Without Entire Matrix Logarithms
- AMR-036-0044 · Partial answer (new negative cases)Open Sets of Linear Systems with n = mp That Are Not Stabilizable by Static Output Feedback: New Cases of a Problem of Eremenko
- KP-1.19 · Affirmative answer (written proof of a recorded result)Asymmetric Hyperbolic Knots Detect Mapping Classes of 3-Manifolds: A Written Proof for a Problem of the K3 List
- AMR-042-0001 · Complete answer (M = 6 for every prime)The Exact Order of Mixing in a Problem of Ward: A Seven-Term Polynomial with No Sparser Multiple
- AIM-TOPOLOGY-0276 · Explicit Witt-family worked exampleA Genus-Two Algebraic Family with Vanishing Real Toledo Numbers and Nonzero Generic Witt Class
- AIM-COMPUTATION-0020 · Sharp conditional-ratio theoremSharp Ambiguity Bounds for Surviving Conditional Odds Ratios
- AIM-COMPUTATION-0095 · Sharp bound and unrestricted-exponent counterexampleA Sharp Obstruction to Uniform Low-Degree Tails over Finite Fields
- AMR-011-0009 · Normal-subgroup intersection counterexampleProperty tau Need Not Survive Intersections of Normal Subgroups
- AMR-103-0120 · Complete counterexample; classification by gauge groupA Conjecture of Guadagnini and Pilo on Quantum Invariants and the Fundamental Group: Counterexamples and a Classification by Gauge Group
- AMR-011-0049 · Complete scoped theoremExact Word-Map Distributions on Four-by-Four Unitriangular Groups
- AIM-PROBABILITY-0093 · Negative answer at the stated scaleUniform Height Bounds for Brownian Excursions with an Exact Area Constraint
- AIM-CONVEX_GEOMETRY-0001 · Minimum-perimeter theoremMinimum Perimeter of Central Hyperplane Sections of the Cube
- AIM-CONVEX_GEOMETRY-0045 · Octahedron minimum-volume theoremMinimum-Volume Central Slabs of the Regular Octahedron
- AIM-OTHER-0011 · Corner-factor theoremCorner Itineraries and Resonance of ASM Superpromotion
- AIM-SEVERAL_COMPLEX_VARIABLES-0028 · Zero-curve theorem and counterexampleAlgebraic Zero Curves of Homogeneous Plurisubharmonic Polynomials
- AIM-COMBINATORICS-0112 · Scoped theorem and orientation counterexampleOrientation Dependence and Exact Groups for Ridge-Routing Sandpiles
- AIM-ARITHMETIC_GEOMETRY-0050 · Scoped scheme-theoretic theoremsFrobenius Factorization Loci for Gauss Maps of Smooth Plane Curves
- AIM-COMBINATORICS-0257 · Scoped lower-bound theoremPolynomial Unit-Distance Lower Bounds over Prime Fields
- PE-1012 · Complete theorems and a new constantParity Rock–Paper–Scissors Games and the Alper Constant
- AIM-PROBABILITY-0013 · Scoped critical-point theoremCritical Points and Local Maxima of Sparse Binary Restricted Boltzmann Machines
- AIM-TOPOLOGY-0043 · Scoped quantum-torsion theoremQuantum Integer Torsion in the Skein Module of a Connected Sum of Two Solid Tori
- AMR-068-0001 · Complete four-point classificationThe Stress-Sign Stratification of Four Labeled Points in Three Dimensions
- AIM-PROBABILITY-0042 · Complete logarithm-removal theoremHitting Moving Targets on Eulerian Digraphs in O(mn) Expected Time
- AIM-COMBINATORICS-0263 · Complete one-point classificationOne-Point Extensions of Multiplicative Subgroups as Self-Sumsets
- AMR-014-0007 · Complete geometric count for (4,2)A Geometric Count of Fourth-Power Decompositions of Binary Octics
- AIM-GEOMETRY-0171 · Complete spherical convergence proofSmooth Convergence of Inverse Gauss Curvature Flow on the Two-Sphere
- KP-4.124 · Computer-assisted theorem (related result; the problem stays open)The Coxeter Group [5,3,3,3] Has No Torsion-Free Subgroup of Index 14400: A Computer-Assisted Result Related to Problem 4.124 of the K3 List
- AIM-TOPOLOGY-0275 · Complete split F4 product-surface theoremThe Quaternionic Toledo Spectrum for Split F4 over Products of Surfaces
- OPG-57403 · Complete answers (a proof and a counterexample)Two Problems of Porton on Reloids: Metamonovalued Reloids Are Monovalued, and S(S(f)) = S(f) Fails
- AIM-GEOMETRY-0038 · Complete finite-type comparison theoremA Common Positive Point for Braid Varieties
- KP-2.28 · Partial answer (explicit classes of graphs; the problem stays open)Sending Loxodromic Elements of Right-Angled Artin Groups to Pseudo-Anosov Mapping Classes: A Partial Answer to Problem 2.28 of the K3 List
- AIM-GEOMETRIC_GROUP_THEORY-0044 · Complete proofSpherical Type Is an Isomorphism Invariant of Artin Groups
- OWR-15427-019 · Complete dense-regime theoremLog-Free Supersaturation of Collinear Triples in Three-Dimensional Grids
- AMR-067-0008 · Partial answer (three further cases; one regime stays open)Volume-Minimising Projective Subspaces in Berger Projective Spaces: Three Further Cases of a Question of Gil-Medrano
- AMR-022-5039 · Partial answer (exact for p ≤ 2; bounds for p > 2)Duren's Problem on Integral Means of the Derivative of a Subordinate Function: the Radius 1/2 for p ≤ 2 and Bounds for p > 2
- AMR-044-0001 · Partial answer (periodicity criterion; periods in special cases)On the Billiard Problem of Dragović and Radnović for Two Concentric Half-Circles: A Periodicity Criterion and the Periods in Special Cases
- AIM-GEOMETRY-0111 · Complete scoped 3D theoremSoft Density Profiles Do Not Determine Vacant Percolation
- AMR-096-0008 · Partial answer (negative answers to two items as literally posed)On Aldous's Question about the Metropolis Chain on Cayley Graphs: Non-Monotonicity of the Relaxation Time
- AIM-GEOMETRY-0112 · Complete lattice theoremExact Small-Softness Stabilization of Euclidean Lattice Packings
- AMR-049 · Seven conjectures decided (four proved, one proved in corrected form, two disproved)On Seven Conjectures of Fuchs on Billiard Trajectories in Regular Polygons and Geodesics on the Regular Dodecahedron
- AIM-OTHER-0067 · Complete crossed-product theoremSharp Tensor Nilpotence Bounds for Finite Crossed Products
- AIM-ANALYSIS-0146 · Complete finite-operator theoremA Gaussian Obstruction to Uniform Mesh Bounds for Differential Operators
- OWR-8415343-001 · Answered: the estimate holds for every smooth G (explicit constants)A Composition Estimate for Fourier Series with the Supremum in Time Inside the Sum, and a Question of Chruściel
- OWR-16415-018 · Partial answer (four-holed spheres, genus 2 only)A Finite Cover of the Genus-Two Surface in Which Every Four-Holed Sphere Has Connected Preimage: A Partial Answer to a Question of D. Gekhtman
- AIM-SEVERAL_COMPLEX_VARIABLES-0021 · Complete counterexampleA Nonremovable Meromorphic Map into a Ball on a Codimension-Two Wedge
- AIM-DYNAMICAL_SYSTEMS-0042 · Complete proof (stated special case)Geometric and Arithmetic Independence of Affine Chebyshev Towers
- AMR-046-0017 · Complete special-case proofA Sharp Three-Cycle Bound for Two-Zone Affine Systems with a Virtual Focus
- AIM-COMBINATORICS-0183 · Complete special-class proofA Sharp Three-Fifths Bound for Seymour Vertices in Three-Regular Oriented Graphs
- OWR-8415347-019 · Answer for every N (no for N = 2, 3; yes for N ≥ 4); nilpotent residual of 𝒪(ON+) determinedThe Free Orthogonal Quantum Group ON+ Is Topologically Generated by SON and the Dual of a Free Group Exactly When N ≥ 4: On a Question of Freslon and the Nilpotent Residual of 𝒪(ON+)
- OWR-16163-011 · Partial answer (structure of the measure; no closed formula)The Law of the Sum of the Two Faces of Bi-Monotone Brownian Motion Has Atoms: A Partial Answer to a Question of M. Gerhold
- AIM-GEOMETRY-0315 · Complete scoped proof and exampleCyclic Vertex Groups and Integral Toric Restriction
- OWR-14299905-036 · Partial answer (two constants determined, with the extremal grids; computer-assisted)The Word-Search Constants C2(ABBB) = 8/5 and C3(ABB) = 6: On Two Questions of Schildkraut and of Halberstam and Schildkraut
- OWR-13494-011 · Complete negative answer (classification theorem)Line Configurations in ℙ3 Whose Symbolic Square Has Initial Degree α(I)+1 Are Coplanar or Pseudostars: A Negative Answer to a Question of Janssen
- OWR-1703876-012 · Complete counterexample for every n ≥ 6; corrected criterionPolygons with Prescribed Turning Angles in Three Dimensions: A Counterexample to a Conjecture of Efrat, Fulek, Kobourov and Tóth, and the Corrected Criterion
- OWR-3385-009 · Complete negative answerThe Graph of a Convex 4-Polytope Is Flat in the Boundary Whenever It Is Linklessly Embeddable: An Answer to a Question of Izmestiev
- AMR-103-0016 · Problem answered in print by others; sharper bound and short proofsOn a Problem of Ferrand and Stoimenow Answered in Print by Rutherford and Ito: A Sharper Signature Bound for the HOMFLY Polynomial of Alternating Links, and a Short Proof that Quasipositive Alternating Links Are Positive
- AIM-ANALYSIS-0055 · Complete fixed-double-pole theoremExact Degrees for a Fixed-Double-Pole Rational Hénon Family
- AMR-036-0017 · Complete counterexample to the proposed alternativesPolynomial-Exponential Counterexamples to a Derivative-Zero Half-Plane Classification
- AMR-036-0023 · Auxiliary counterexample and conditional theoremsRadial Barriers for Discrete Newtonian Potentials: Counterexamples and Sufficient Conditions
- AIM-TOPOLOGY-0097 · Coxeter higher-A vanishing theoremHigher A-Homotopy Vanishing for Coxeter Chamber Graphs
- AIM-GEOMETRIC_GROUP_THEORY-0042 · Sharp multipartite spectral theoremSharp Degree Bounds for Multipartite Lorentzian Coxeter Forms
- OWR-12697693-003 · Complete negative answer (already for left braces)The Elements with Property (s) of a Skew Brace Need Not Form an Ideal: On a Question of Colazzo, Ferrara and Trombetti
- AMR-022-7031 · Complete answer: comparison theorem and characterizationOn Hayman's Problem on the Series Σ f(cn/an) Formed with Second Partial Sums: A Comparison Theorem and a Characterization
- AIM-GEOMETRY-0201 · Singular example and minimality theoremExpansive Symmetry Breaking at a Five-Vertex Linkage Singularity
- AIM-PROBABILITY-0118 · Attributed continuation theoremSymmetric Unimodality for Real Free Convolution Powers
- AIM-PROBABILITY-0053 · Complete weighted Gibbs theoremTotal Variation Mixing of Weighted Quadratic Gibbs Samplers on a Cube
- OWR-15993-009 · Five steady states for chain lengths N ≥ 651 (computer-assisted); at most three for N ≤ 4Five Steady States in the Agonist-Only Rendall–Sontag Model for the Initiation of T Cell Activation: Certified Examples with Chain Lengths from 651 to 4000
- OWR-12697710-006 · Explicit counterexamples in five and in seven variables (the first negative answer is due to a prior preprint)On Reznick's Question about the Cone of Forms with a Sum-of-Squares Odd Power: Explicit Non-Convexity for Sextics in Five and Quartics in Seven Variables
- OWR-16164-020 · Complete proof: divisibility by the Catalan power of det A for every nOn a Determinant of Teufl and Wagner Related to Set Partitions: Divisibility by the Catalan Power of det A
- AIM-PROBABILITY-0028 · Complete scoped counterexampleLinear Gaussian SURE Bias for Haar Soft Thresholding under Smooth Noise
- OWR-793-006 · Complete negative answer (an explicit primitive group of twisted wreath type)A Negative Answer to a Question of Praeger and Seress on Minimal Subdegrees of Primitive Groups of Twisted Wreath Type
- AMR-029-0044 · Complete proof: the path condition suffices for all roots and all convex setsIndependent Arborescences in Acyclic Digraphs: The Path Condition Suffices for All Roots and All Convex Sets
- AMR-109-0082 · Observation: negative answer (unbounded multiplicity for every term of the Johnson filtration)Unbounded Multiplicity in the Dilatation Spectrum of Every Term of the Johnson Filtration: An Observation on a Question of Farb
- AIM-ANALYSIS-0041 · Complete scoped theoremsFinite-Dimensional Lp-Bergman Spaces on Complete Reinhardt Domains
- AIM-ANALYSIS-0056 · Complete scoped criteriaSharp Attraction Thresholds for Neutral Jordan Skew Products
- AIM-PHYSICS-0055 · Complete planar theoremConstant Normalized Area in the Planar Three-Body Problem
- OWR-4990374-015 · Complete proof of the conjecture in every rank (characteristic zero)Free Hyperplane Arrangements with Exponents 1, 2 and at Most One 3 Are Supersolvable: A Proof of a Conjecture of Tohăneanu
- AIM-ALGEBRAIC_GEOMETRY-0211 · Complete scoped toric theoremA Pivot-Independent Toric Model for Framed Carlsson-Weinshall Duality
- AMR-020-0202 · Complete scoped counterexampleGlobal Symplectic Equivalence Does Not Determine a Lagrangian Torus Fibration
- AMR-011-0031 · Complete counterexampleA Cayley Graph with Chromatic Number Three and No Invariant Random Three-Coloring
- AMR-050-0043 · Complete scoped proofCosine Sums of Focal-Inversive Elliptic Billiards
- OWR-741-001 · Complete counterexample from rank 6 on; proof up to rank 5Generalized Hive Polytopes Have Integral Points Exactly up to Rank 5: A Counterexample to a Conjecture of De Loera and McAllister
- AIM-COMBINATORICS-0007 · Complete quantitative counterexampleA Counterexample to a Linear-Junta Inverse Theorem for Spherical Triangles
- AIM-PROBABILITY-0086 · Complete fixed-parameter proofSpatial Mollification of the Multilayer Stochastic Heat Equation
- AIM-GEOMETRIC_GROUP_THEORY-0059 · Complete scoped corollaryLargest Acylindrical Actions for Artin Groups under a Strict Triple Condition
- AIM-SEVERAL_COMPLEX_VARIABLES-0011 · Complete scoped theoremLower Bounds for Positive-Power Exponents in a Hermitian Polynomial Family
- AIM-ALGEBRAIC_NUMBER_THEORY-0077 · Complete scoped corollarySemisimple Torsion Fields of Modular Jacobians at Powers of 11 and 13
- AIM-ANALYSIS-0148 · Complete scoped theorem and counterexampleEventual Scalar Recurrences for Constrained Orthogonal Polynomials
- AIM-TOPOLOGY-0101 · Complete scoped nonuniqueness counterexampleBinary Interior Excision and Nonuniqueness of Graph Homology
- AMR-040-0001 · Complete scoped counterexample and theoremEuclidean Realizability of Fundamental-Polytope Types
- OWR-4136-005 · Complete construction-specific counterexampleRademacher's Half-Ball and Cone Counterexample in Three Dimensions
- OWR-15957-003 · Complete scoped theoremThe Maximal Finite CP-Plus-Rank Equals the Maximal CP-Rank
- OWR-15221-002 · Complete scoped theoremA Sharp Prior-Sensitivity Threshold for Fixed-Partition Support Boundaries
- OWR-1111-001 · Complete answer: NP-hard, with and without edge costsBlocking Shortest Paths between Three Terminals Is NP-Hard: A Reduction from Three-Terminal Cut
- AIM-ANALYSIS-0095 · Complete scoped theoremCoordinate Subcubes of Weaver Partitions at the MSS Bound
- AIM-PROBABILITY-0133 · Complete scoped counterexampleGeometric Sampling Does Not Determine Regularized R-Diagonal Laws
- AIM-GEOMETRIC_GROUP_THEORY-0126 · Complete scoped proofAn Explicit Generating Pair for Radu’s Simple Tree Lattice
- AIM-GEOMETRY-0095 · Complete scoped proofEventual Square Pyramid Optimality for Five-Point Riesz Energy
- AMR-109-0020 · Negative answer to the example question: not uniqueRiemann Surfaces of Genus 9 with Nilpotent Automorphism Group of the Largest Order Are Not Unique: On a Question in Farb's Problem 2.19
- OWR-13494-009 · Containment proved for every n = 3m (cluster reading)Containment of the Symbolic Cube in the Square for the Roulleau–Urzúa Configurations of Cubics
- AMR-105-0051 · Complete answer: yes, explicit examplesFree Knots and Links with Two Minimal Representatives: Explicit Examples for a Problem of Fenn, Ilyutko, Kauffman and Manturov
- OWR-793-006 · Negative answer: an infinite family attains the boundAn Infinite Family of Primitive Groups of Twisted Wreath Type Attaining the Upper Bound for the Minimal Subdegree
- OWR-1738-011 · Complete answer: both bounded, by 4A Uniform Bound for Free Pairs of Bicyclic Units and for Free Points at Algebraic Integers
- AMR-014-0004 · Counterexamples: the conjecture is falseCounterexamples to a Conjecture of Fröberg, Lundqvist, Oneto and Shapiro on Maximal k-Ranks of Binary Forms
- AMR-099-0065 · Partial answer: the recursive constant exceeds the BRW constantOn a Question of Benjamini on First-Passage Percolation in Hierarchical Graphs: The Recursive Constant Exceeds the Branching Random Walk Constant
- AMR-096-0023 · Part (a) proved; parts (b), (c) false in a parameter rangeOn Conjecture 1 of Aldous for a Spatial Model of City Growth: A Proof of Part (a), and Counterexamples to Parts (b) and (c) for 2α/√3 < β < 2α
- AMR-005-0008 · Complete proofExplicit Rotation and Arithmetic Obstructions for Null Chains on a Lorentz Ellipsoid
- AIM-PROBABILITY-0113 · Complete proof of the CAR subquestionSharp Norms of Rectangular Majorana Matrices and a Two-Matrix Obstruction to Strong Convergence
- AIM-TOPOLOGY-0147 · Complete proof for the stated entropy modelThe Entropy Cost of Infinite-Genus Weil-Petersson Local Limits
- OWR-15587-006 · Counterexamples: lattice hulls of balls need not be normalLattice Hulls of Euclidean Balls Need Not Be Normal: Counterexamples in Every Dimension at Least Six
- OWR-723-005 · Negative answer for the expected p-th powerA Negative Answer to a Question of Novak on Randomly Shifted Lattices and the Expected p-th Power of the Lp Discrepancy
- OWR-13677-003 · Complete answer: yes, even co-meagerPurely Singular Copulas Are Typical for the Markov-Kernel Metric D1
- OWR-14299292-013 · Answer: yes, an elementary invariant separates the pairOn a Question of Stanojkovski: Abelian Subgroups of Breadth Two Distinguish Vaughan-Lee's Groups 8.5.14 and 8.5.15
- AIM-ALGEBRAIC_GEOMETRY-0078 · Complete proof of ambient multiplicity formulaGeneric Multiplicities of Hilbert Schemes on Thickened Lines
- OWR-14601-005 · Complete answer: the cone is GorensteinThe Affine Cone over the Second Secant Variety of (ℙ1)5 Is Gorenstein: On Three Questions of Michałek
- OWR-4428-011 · Partial answer: irrational algebraic valuesIrrational Algebraic Values of Grabowski's Automaton Series and an Irrational Algebraic ℓ²-Betti Number of the Lamplighter Group
- AIM-COMPUTATION-0021 · Complete proof of scoped augmentation resultsMinimum Support Augmentation in Chordal Graphical Models
- AIM-ALGEBRAIC_GEOMETRY-0190 · Complete proof of scoped calibration theoremExact Positive-Focal Certificates for Fundamental-Matrix Pencils
- AIM-TOPOLOGY-0106 · Complete fixed-test classificationInduced-Cover Excision Characterizes Dismantlable Hom Tests
- AIM-FUNCTIONAL_ANALYSIS-0012 · Complete finite-frequency theoremFinite-Frequency Rigidity for Maximal Perimeter Sections
- OWR-4428-012 · Partial answer: the square root of two is an ℓ²-Betti numberThe Square Root of Two Is an ℓ²-Betti Number Arising from the Lamplighter Group
- AIM-ALGEBRAIC_GEOMETRY-0035 · Complete external obstructionA Rank-One Obstruction for Odd-Primary Wood Equivalences
- AMR-103-0231 · Complete counterexampleA Torsion-Rank Obstruction to Turaev's Alexander Polynomial Realization Conjecture
- AIM-ALGEBRAIC_GEOMETRY-0184 · Complete scoped theoremFour- and Five-View Thresholds for Calibration with Known Focal Scales
- OWR-14298804-012 · Partial answer: fare zones under bounded congestion, settled up to polylog(k)Fare Zone Assignment with Hard Cut Budgets under Bounded Congestion
- OWR-4428-012-B · Partial answer: genuine witnesses and irrational values for every pGenuine Witnesses for Virtual ℓ²-Betti Numbers of Ara and Claramunt and Irrational Algebraic ℓ²-Betti Numbers of ℤ/p ≀ ℤ
- OWR-14299904-015 · Complete answer: the maximum is determined for all n, k, dOptimizing over Nonzero Littlewood–Richardson Coefficients: The Quantum Max d-Cut of Complete Bipartite Graphs
- OWR-15215-006 · Partial answer: the optimal constant grows exponentiallyOn Meckes's Monotonicity Question for the Expected Volume of a Random Simplex: The Optimal Constant Grows Exponentially
- AIM-OTHER-0057 · Complete cyclic-family theoremBrauer-Picard Groups and Module Orbits for Twisted Cyclic Categories
- EP-963 · Partial answer: the conjecture holds for all n up to 31Every Seven Positive Reals Contain a Dissociated Four-Element Subset: Small Cases of an Erdős Problem on Dissociated Sets
- AMR-050-SYM7 · Complete answer: seven identities proved, exceptional sets statedCentral Symmetry and Seven Area Identities for Even N-Periodics in the Elliptic Billiard
- OWR-9790358-011 · Partial answer: an explicit rule for n = 3, no symmetric rule for n = 4Gaussian Cubature of Even Degree for the Product Chebyshev Weight in Three Variables: An Explicit Ten-Node Rule and No Twenty-Node Rule with a Sign Symmetry
- OWR-15986-005 · Negative answer to Question 1, by a short argumentOn a Question of Becher and Slaman: For Kurtz Random x the Discrepancy of (2nx) Is Not O((log N)/N)
- AMR-050-0011 · Complete answer: three pedal-area identities for every pointPedal Areas of Periodic Elliptic Billiard Trajectories with Respect to an Arbitrary Point
- OWR-11695865-007 · Complete tree singular-locus theoremAn Exact Singular-Locus Formula for Rooted-Tree Compartmental Models
- AIM-ANALYSIS-0032 · Complete convex sharp-constant theoremConvex Near-Extremizers for the Multidimensional Hermite-Hadamard Inequality
- AMR-050-0018 · Disproof as printed; the focal form of the row is provedAntipedal Areas of N-Periodics in the Elliptic Billiard: Row k401 Fails at the Centre and Holds at the Foci
- AIM-ANALYSIS-0019 · Complete weak-coupling theoremUniform Weak Robin Comparisons on Triangles and a Cubic Moment Formula
- AIM-ANALYSIS-0016 · Complete explicit counterexampleA Four-Point Obstruction to Completeness for Three Atomic Inner Functions
- AMR-050-0022 · Complete answer: five focal area identities provedFive Focal Area Invariants of N-Periodics in the Elliptic Billiard
- AMR-050-0056 · Complete answer: five inversive-object rows decidedPolar, Dual and Inversive Polygon Areas of N-Periodics in the Elliptic Billiard
- AMR-050-0049 · Complete answer: four inversive area rows proved for every NMidpoint Polygons and Four Inversive Area Invariants of N-Periodics in the Elliptic Billiard
- AIM-ALGEBRAIC_GEOMETRY-0271 · Complete scoped proofSharp Risk Bounds and a Shrinking-Variance Transition for Normal and Gamma Fits
- AMR-050-0002 · Corrected form proved; two rows proved on their domainsHalf-Angle Sines, Steiner Pedal Areas and Evolute Areas of N-Periodics in the Elliptic Billiard
- AIM-CONVEX_GEOMETRY-0019 · Complete proof (classical corollary)A Geometric-Mean Proof of the Spherical Simplex Volume-Product Bound
- AIM-GEOMETRY-0388 · Complete theorem (product family)Paneitz Resonances and Index Growth on a Sphere Times a Hyperbolic Surface
- OPG-59927 · Partial answer: exact values for K3,b with b at most 8On the Ohba Number of K3,b: Exact Values for b ≤ 8 by a Certified Exhaustive Search
- OWR-15988-004 · Complete theorem (two-dimensional case)The Scaling-Rotation Chain Metric for Positive-Definite Two by Two Matrices
- AIM-COMBINATORICS-0153 · Complete theorem (recursive construction)An Exact Logarithmic-State Algorithm for Recursive Tight-Cycle Constructions
- OWR-14298158-012 · Negative answer: a proper subclass of the separable permutations whose generating function lies outside the fieldA Proper Subclass of the Separable Permutations Whose Generating Function Lies Outside ℚ(z, √(1−4z), √(1−6z+5z²))
- AMR-022-4023 · Partial answer: connected lemniscates in every dimension lie in a ball of radius 2On a Question of Erdős on Lemniscates in ℝm: Connected Lemniscates Lie in a Ball of Radius 2
- AIM-REPRESENTATION_THEORY-0033 · Complete theorems (specified models)Central Obstructions and Block-Complement Categories for Schur Covers
- OWR-3385-011 · Partial answer: a star-shaped spherical heptagon whose development self-intersectsA Star-Shaped Spherical Heptagon Whose Development Self-Intersects
- AMR-014-0005 · Complete classification (cube rank)Cube Ranks of Monomials
- AIM-COMBINATORICS-0085 · Complete theorem (layered colorings)Essential Periods and Monochromatic Triangles in Measurable Layered Colorings
- OPG-369 · Complete theorem (four weight classes)The Schrijver-Seymour Bound for Four Weight Classes
- AMR-096-0036 · Solved: S(1) of a scale-invariant random spatial network is not a treeThe Subnetwork S(1) of a Scale-Invariant Random Spatial Network Is Not a Tree
- AMR-020-0402 · Solved: the algebraic Birkhoff conjecture for outer billiards on the sphere and the hyperbolic planeOn the Algebraic Birkhoff Conjecture for Outer Billiards on the Sphere and the Hyperbolic Plane: A Corollary of Theorems of Glutsyuk
- AMR-051-0010 · Complete theorem (smooth maps and flows)Full-Dimensional One-Sided Trapping for Smooth Area-Preserving Plane Maps
- OPG-37656 · Complete proof (K5 concentration)Three-Colourability of Random Lifts of K5
- OWR-14299911-020 · Partial answer: the travelling polytope constant of every product of treesThe Travelling Polytope Constant of a Product of Trees: An Answer to a Question of Genevois
- OWR-1782-017 · Partial answer: real-rooted integer polynomials satisfy a g-theorem (Conjecture 2 proved)Real-Rooted Integer Polynomials Satisfy a g-Theorem: Proof of a Conjecture of Brenti and Welker
- OWR-12007-010 · Partial answer: no nontrivial word has vanishing rotation number under every circle actionElliptic Values of Word Maps on SL(2,ℝ) and a Question of Calegari and Walker on Rotation Numbers
- OWR-4531-016 · Partial answer: the cubic Fermat equation has nontrivial solutions in some model spacesThe Cubic Fermat Equation in Model Spaces: A Negative Answer to a Question of Dyakonov
- AIM-GEOMETRY-0110 · Complete scoped proof (finite stability)Saturated Aperiodic Packings with Finite Stability
- AMR-020-0507 · Solved: quadratic Poisson brackets without polynomial integrable systemsOn a Question of Bolsinov, Matveev, Miranda and Tabachnikov: Quadratic Poisson Brackets without Polynomial Integrable Systems
- GRAPH-049 · Complete connected follow-up proof with induced minimalityNon-Word-Representable Single-Triangle Graphs with Word-Representable Line Graphs
- AIM-ARITHMETIC_GEOMETRY-0079 · Complete equidegree-family theoremSharp Interpolation for Equidegree Hilbert–Burch Curves
- OPG-164 · Complete spider reconstruction theoremSharp Graham-Sequence Reconstruction for Spiders
- AMR-021-0006 · Complete separated-system theoremSharp Finite Zero Bounds for One-Parameter Separated Polynomial Systems
- OWR-720-011 · Solved: the canonical normalized tight frame of a wavelet frame need not have wavelet structureThe Canonical Normalized Tight Frame of a Wavelet Frame Need Not Have Wavelet Structure: An Explicit Riesz Wavelet for a Question of Han and Frank
- AIM-SEVERAL_COMPLEX_VARIABLES-0026 · Complete sharp polynomial-kernel theoremSharp Positivity Tests for Polynomial Hermitian Kernels
- EP-668 · Complete finite congruence classificationTwo Congruence Classes of Extremal Twenty-Point Sets
- AIM-PROBABILITY-0007 · Complete scoped counterexamplesFinite Likelihood Traps on Connected Boltzmann Trees
- AIM-PROBABILITY-0103 · Complete scoped counterexamplesHidden Constants after Free Semicircular Regularization
- OWR-9790365-007 · Complete proof of question 1A Sharp Gaussian Log-Partition Bound for Mutual Information
- AIM-SEVERAL_COMPLEX_VARIABLES-0020 · Boundary counterexample and jet theoremInfinite Boundary-Jet Ambiguity for Proper Embeddings into a Smooth Strongly Convex Domain
- OWR-15585-006 · Complete affirmative proofStrong Approximate Innerness for Diagonal Graph Actions without Aperiodicity
- AIM-COMPUTATION-0040 · Explicit theorem and sharp boundA Connected Positive Model with a Nonsemialgebraic Likelihood Cell
- AIM-COMBINATORICS-0181 · Complete counterexampleCyclic Counterexamples to the Exact-Distance Higher-Neighborhood Conjecture
- AMR-122-0012 · Complete proof for every nOn Propp's Problem 12: The Sum of the Entries of the Inverse Kasteleyn–Percus Matrix of the Aztec Diamond
- KOU-16.60 · Complete affirmative answerElements Inverted by an Automorphism and the Sum of the Character Degrees (Kourovka Notebook Problem 16.60)
- AMR-126-0025 · Negative answer (sequence reading)Degree Sequences of Rational Maps Need Not Be Piecewise Linear Recurrent
- AMR-152-0049 · Conjecture false for every rational JacobianQuadratic Hénon Maps over ℚ with Rational Cycles of Length Three and Four for Every Jacobian
- KOU-13.19 · Complete answer: regular iff n ≤ pQuotients of Subdirect Products and Regular p-Groups (Kourovka Notebook Problem 13.19)
- AMR-150-0035 · Complete answer: exactly the left-Σ2 realsHausdorff Dimensions of Computable Closed Subgroups of Finitely Generated Profinite Groups
- KOU-7.27 · Affirmative answer for all large qLevchuk's Problem on Diagonal Matrices in SLn(q) (Kourovka Notebook 7.27): An Affirmative Answer as a Corollary of Theorems of Fulman and Guralnick
- AIM-GEOMETRIC_GROUP_THEORY-0024 · Complete proof (frozen ab formulation)A Compact Cubulation of a Quadratic Free-by-Cyclic Group
- AIM-OTHER-0095 · Complete normalization obstructionWhen Binary-Form Norm Balls Have Finite Haar Volume
- AIM-ALGEBRAIC_GEOMETRY-0162 · Complete scoped counterexampleInfinitesimal Lifting Does Not Imply Flatness for Green Functors
- AIM-CONVEX_GEOMETRY-0026 · Complete scoped counterexampleSubgaussian Convex Bodies Need Not Have Subgaussian Projections
- AIM-GEOMETRIC_GROUP_THEORY-0023 · Complete family classificationA Cycle Criterion for CAT(0) Signed Permutation Mapping Tori
- AIM-GEOMETRY-0093 · Complete lattice theoremEntropy Selection and Shape Drift for Gaussian Lattice Minima
- OWR-15951-003 · Partial answer: yes for constant bumps and all couplings; conditional result for bumps of bounded variationOn Fillman's Question for Separable Continuum Fibonacci Operators: Half-Lines for Two Couplings and the High-Energy Fricke–Vogt Invariant
- AMR-155-0008 · Affirmative answer for every b in (1/8, 1/6]The Functions ψb of Bownik and Weber Are Frame Wavelets for 1/8 < b ≤ 1/6
- AMR-154-0006 · Explicit regular conservation laws for every normal form; all real-analytic laws describedExplicit Conservation Laws for the Normal Forms of Two-Dimensional gl-Regular Nijenhuis Operators
- AMR-129-0019 · Partial answer: dense in balls (source not the centre) and in ellipses (source not a focus)On Tabachnikov's Question on the Density of Reflected Wave Fronts: Balls and Ellipses
- AIM-ALGEBRAIC_GEOMETRY-0318 · Complete projection counterexampleA Nodal Decic Counterexample to the Generic Projection Incidence Bound
- AMR-074-0117 · Answer depends on the reading: no under the null-cover reading (RCA0 proves LDT)The Lebesgue Differentiation Theorem over RCA0: On Question 17 of Montalbán's Open Questions in Reverse Mathematics
- OWR-12339-005 · Unit coefficients under weak nondegeneracyUnit-Coefficient Simplex Decompositions under Weak Nondegeneracy
- OWR-17464-019 · Complete degree-two bound for all uniformitiesIndependent Sets in Two-Regular Uniform Hypergraphs
- OWR-2090-028 · Complete measure-zero theoremCommon Support Lines of Three Disjoint Convex Sets Have Zero Volume
- OWR-13941-002 · Complete convexity counterexampleNonconvexity of the First Artin Cone in Six Variables
- AIM-COMPUTATION-0052 · Explicit polynomial certificateAn Explicit Nullstellensatz Certificate for the Seventh Hadamard Ideal
- OWR-2090-019 · Complete negative answerUnbounded Line Pinning by Pairwise Disjoint Convex Bodies
- OWR-16412-006 · Complete counterexample to Question 5A Bounded Convex Hilbert Domain with Vanishing Squeezing
- AIM-PHYSICS-0025 · Complete connected counterexampleMore Colors Can Slow Mixing on a Four-Vertex Path
- OWR-16930-001 · Complete canonical percolation proofHigh-Density Canonical Hard Disks Percolate
- AMR-029-0087 · Complete banana-tree algorithmDivisorial Gonality of Banana Trees in Quadratic Time
- AIM-COMBINATORICS-0131 · Complete kernel theoremCritical Groups Detect Homology Actions of Simple Graphs
- OWR-13490-013 · Counterexample and repaired theoremComponent Symmetry and Tightness in Dimension Two
- OWR-1787-006 · No: polynomial multiples of MacLane universal functions need not be universalPolynomial Multiples of MacLane's Universal Functions Need Not Be Universal: A Negative Answer to a Question of Costakis
- OWR-1315-001 · Complete proofProjective Units and Witt Vanishing for Stable Module Categories
- OWR-12868-001 · Complete calculation for explicit familiesCohomology Annihilators of Truncated Polynomial Rings
- AMR-103-0230 · Partial answer: new infinite families and all open determinants up to 106Determinants of Prime Alternating Achiral Knots: A Partial Answer to a Problem of Stoimenow
- OPG-60013 · Partial answer: exact values for n up to 13 and linear boundsWeak Saturation of the Cube in the Clique: Exact Values for n ≤ 13 and Linear Bounds
- OWR-11129-006 · Yes: order 1/log n, with a log-periodic asymptoticElements Powering to Strong Involutions in Finite Classical Groups of Odd Characteristic: Order 1/log n and a Log-Periodic Asymptotic
- OWR-1061-006 · Partial answer: non-regenerative for three classes and off one mutation rateNon-Regenerative Sampling Distributions of Ξ-Coalescents: Möhle's Conjecture with a Kingman Component, for Finitely Many Atoms or Boundedly Many Dust-Free Boxes, and Off One Mutation Rate
- AMR-123-0003 · Partial answer: an infinite family and all open cases up to max(a,b) = 60A Partial Answer to Propp's Problem 7 on Benzel Tilings: An Infinite Family of Type-103 Tilings and the Open Cases with |a−b| ≤ 7 or max(a,b) ≤ 60
- OWR-14299288-010 · Complete scoped sampling theoremA Sharp Sampling Bound for Primitive Real Toric Arrangements
- OWR-3392-009 · Complete ambient-invariance counterexampleAmbient Dependence and Extreme Extension for Exchangeable Free-Product States
- AMR-119-0003 · CR or anti-CR: no as literally posed, yes for coorientation-preserving mapsConformal Homeomorphisms of Strongly Pseudoconvex CR 3-Manifolds Are CR or Anti-CR: On a Question of Puqi Tang
- OWR-4530-006 · Yes: the two-chunk DFC system is stable although its large-system limit is notPositive Recurrence of the Two-Chunk Deterministic First Chunk System: A Stable Population Process with an Unstable Large System Limit
- OWR-17469-002 · The limit exists and depends on the domain only through its volumeThe Asymptotic Constant for Optimal Integration of Ck Functions on Bounded Lipschitz Domains: Existence and Dependence on the Volume Only
- OWR-4425-012 · No in the universal reading; fixed alphabets classified up to one small windowWhen Can Patterns Characterise Fractional Powers? On a Question of Shallit
- OWR-12339-003 · Yes: the Gaussian Gabor system over Lambda3 is a frame (computer-assisted)The Gaussian Gabor System over the Complex Lattice Λ₃ of Gröchenig and Lyubarskii Is a Frame: A Computer-Assisted Proof
- OWR-13498-009 · No for nonconvex polyhedra: an explicit pair of triangulated bipyramidsDihedral Angles Do Not Determine Face Angles of Nonconvex Polyhedra: An Explicit Pair of Triangulated Bipyramids
- AIM-PROBABILITY-0016 · Complete scoped heat-flow theoremHeat Flow Leaves Two-Hidden-Unit Restricted Boltzmann Models
- AMR-109-0231 · Yes, as a promise problem: an algorithm that outputs Mosher's exponent MAn Algorithm for the Exponent in Mosher's Theorem on Schottky Subgroups of Mapping Class Groups
- OWR-12872-018 · Proved for every corner weight a ≥ 0; the range −1 to 0 stays openOn the Albert–Bousquet-Mélou Conjecture for Corners in Quadrant Loops: The Case of Nonnegative Corner Weight
- AMR-021-0002 · Complete rational-exponent theoremFinite Equilibria for Rational Inverse-Power Fields in Any Dimension
- OWR-12978-004 · No for group algebras: odd-order Brauer pairsOn Montgomery's Question on Frobenius–Schur Indicators and Gauge Equivalence: A Negative Answer for Group Algebras
- OWR-13498-012 · Yes at generic positions, for every graph with 2n−2 edgesRote's Curious Identity on Self-Stresses: Independence of the Two Points for All Graphs, Existence at Generic Positions, and a Volume Interpretation
- OWR-4990373-005 · False as stated: explicit finite graphs with exact certificatesFinite Counterexamples to the Monotonicity Conjecture of Amir, Baldasso and Beilin for Median and Majority Dynamics
- OWR-14213-008 · False for n = 8, 9 and 11Karpenko's Conjecture on the Special Schubert Class in the Gamma Filtration Fails for n = 8, 9 and 11: A Short Consequence of Printed Results
- OWR-17294-007 · Complete orbit-lattice calculationOrbit Lattice Formulas for Second Homology of Finite Abelian Quandles
- OWR-15219-013 · No: one symmetric cubic refutes Conjectures 2 and 5A Symmetric Counterexample to Brändén's Majorization Conjectures for Mixed Characteristic Polynomials
- AIM-PROBABILITY-0153 · Complete H3 calculationHomotopy Types and Shelling Certificates for the H3 Absolute Order
- OWR-14299097-014 · Complete unitary-family theoremLinear-Length Mixed Identities for Finite Projective Unitary Groups
- AMR-022-7023 · No in the range asked: sufficient exactly when q is below 1/3On Krzyż's Problem 7.23 in Hayman and Lingham: Kühnau's Two-Point Condition Implies Quasiconformal Extendability If and Only If q < 1/3
- OWR-14299580-006 · No: Aff(ℤ/5,2) and Aff(ℤ/5,3) are twist-related but not isomorphicDrinfeld Twists Between Non-Isomorphic Rack Solutions: A Negative Answer to a Question of Ferri
- OWR-17471-010 · Complete counterexampleZero Surgery Obstructs Persistent Free Subgroups of the Figure-Eight Knot Group
- OWR-17288-004 · No in the same-ℝN reading, for every N; flat tori fail the hypothesisLoose Embeddings in a Fixed Euclidean Space: Countable Counterexamples and Flat Tori
- OWR-14299292-011 · Partial answer: yes for p = 5; no for p = 7, 13, 19; open from p = 31 onFixed-Point-Free Representations of Alternating Groups of Twin-Prime Degree: A Partial Answer to a Question of M. Lee
- KOU-19.93 · No for p = 2, 3, 5: part b) of Spiga's conjecture is falseOn Part b) of Kourovka Notebook Problem 19.93: Finite p-Groups with Unbounded Lower p-Central Factors and No C_p ≀ C_p Quotient for p = 2, 3, 5
- KOU-15.98 · No: a counterexample of order 2304, the least possibleMinimal Non-F-Groups with Sylow Residual and the Unitary Groups PSU(3, p^{2n}): A Negative Answer to Kourovka Notebook Problem 15.98
- KOU-2.78 · Partial answer: both sets are finite; the second question stays openOn Trofimov's Group-Theoretic Numbers (Kourovka Notebook Problem 2.78): The Soluble and the Absolutely Simple Numbers Form Finite Sets
- KOU-19.82 · Partial answer: yes for chief factors of type PSL(2,q) and Sz(q), with one restrictionOn Shumyatsky's Question on Soluble Subgroups and the Generalized Fitting Height: A Partial Answer for Groups with Chief Factors of Type PSL(2,q) and Sz(q)
- KOU-11.16 · Partial answer to part b): a reduction to the Suzuki groupsOn Brandl's Question on Finite Simple Groups of Bounded Engel Depth: A Reduction to the Suzuki Groups
- KOU-17.16 · Partial answer: linear except possibly for six exceptional typesOn Bardakov's Question on the Linearity of Automorphism Groups of Artin Groups of Finite Type: A Reduction to Six Exceptional Types
- KOU-15.75 · Partial answer: two infinite families of PSL(2,p) and finite certificatesOn Plotkin's Question on Two-Variable Identities Characterizing Finite Soluble Groups: Two Infinite Families of PSL(2,p) and Finite Certificates
- KOU-11.18 · Partial answers: seven infinite families of G(a,b); SL(2,K) images reduce to G(d,d)Partial Answers to Kourovka Problems 11.18 and 17.80: Infinite Groups G(a,b) and Their Two-Dimensional Images
All Problems
LV0
Problem 1
Multiples of 3 and 5
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Problem 2
Even Fibonacci Numbers
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Problem 3
Largest Prime Factor
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Problem 4
Largest Palindrome Product
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Problem 5
Smallest Multiple
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Java
LV0
Problem 6
Sum Square Difference
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Python
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LV0
Problem 7
10001st Prime
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LV0
Problem 8
Largest Product in a Series
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Java
LV0
Problem 9
Special Pythagorean Triplet
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Problem 10
Summation of Primes
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LV2
Problem 11
Largest Product in a Grid
C++
Python
Java
LV1
Problem 12
Highly Divisible Triangular Number
C++
Python
Java
LV1
Problem 13
Large Sum
C++
Python
Java
LV1
Problem 14
Longest Collatz Sequence
C++
Python
Java
LV2
Problem 15
Lattice Paths
C++
Python
Java
LV0
Problem 16
Power Digit Sum
C++
Python
Java
LV2
Problem 17
Number Letter Counts
C++
Python
Java
LV2
Problem 18
Maximum Path Sum I
C++
Python
Java
LV2
Problem 19
Counting Sundays
C++
Python
Java
LV0
Problem 20
Factorial Digit Sum
C++
Python
Java
LV1
Problem 21
Amicable Numbers
C++
Python
Java
LV0
Problem 22
Names Scores
C++
Python
Java
LV3
Problem 23
Non-abundant Sums
C++
Python
Java
LV2
Problem 24
Lexicographic Permutations
C++
Python
Java
LV1
Problem 25
1000-digit Fibonacci Number
C++
Python
Java
LV3
Problem 26
Reciprocal Cycles
C++
Python
Java
LV1
Problem 27
Quadratic Primes
C++
Python
Java
LV0
Problem 28
Number Spiral Diagonals
C++
Python
Java
LV1
Problem 29
Distinct Powers
C++
Python
Java
LV0
Problem 30
Digit Fifth Powers
C++
Python
Java
LV2
Problem 31
Coin Sums
C++
Python
Java
LV1
Problem 32
Pandigital Products
C++
Python
Java
LV0
Problem 33
Digit Cancelling Fractions
C++
Python
Java
LV0
Problem 34
Digit Factorials
C++
Python
Java
LV1
Problem 35
Circular Primes
C++
Python
Java
LV0
Problem 36
Double-base Palindromes
C++
Python
Java