Open research questions in Mathematics and Applications
45 unresolved questions extracted from the limitations and future-work sections of 1,061 Mathematics and Applications papers in our library. Each links back to the study that raised it.
What the literature leaves open
The individual divisibility result is a known consequence of standard digit-sum congruences, but the systematic organization of the full family of inversion differences is new.
To study the number of almost ordinary triangles in a set of points in higher dimensions. To develop more efficient algorithms for counting all c-ordinary triangles. To apply the technique for distinguishing between two cases to other geometric problems.
The paper identifies a gap in the understanding of the incidence results that involve points of tangency between the circles in the construction. The paper identifies a gap in the study of Steiner chains and their relation to conic structures.
The need for a deterministic parameterization of a classical spatial envelope. The lack of a compact representation for reconstruction of the Matrix boundary geometry, planar bounding envelope, and three-dimensional volume envelope.
The Pythagorean theorem does not explain why a rope of fixed length pinned at three points must form a specific triangle. The theorem does not provide a constructional framework for deriving the right triangle family.
Further exploration of digit-reversal operations in other bases. Investigation of iterated inversion sequences.
The number of ordinary triangles in a set of points in the plane is not well understood. The problem of finding almost ordinary triangles in a set of points has not been extensively studied.
To generalize results of Becker and Dumas. To provide a new construction of weighted automata that generate classical Z-regular sequences. To address the gap between the results for isolating and non-isolating Z-Mahler equations.
The paper does not provide a complete characterization of automatic sequences. The gap between the results for isolating and non-isolating Z-Mahler equations is not fully addressed.
There was a gap in proving a conjecture of Aluffi. The relationship between the adjoint polynomial and the covolume polynomial was not fully understood.
The lack of a minimal commutative foundation for paired complex numbers. The need for a symmetric system of two planes, two angles, and two angle differences.
Extension of the helical form to predicted elements and physics past Z = 118. Application of the topological framework to other fields, such as materials science.
The Helical Periodic Table: The Pre-Mendeleev 3D Origin, Rubber-Sheet Pliability, and the C₆₀ as One Admissible Host Among Many · 2026 · DOIThe periodic table's natural shape and topology are not well understood. The relationships between the table's elements are not fully captured by traditional 2D representations.
The Helical Periodic Table: The Pre-Mendeleev 3D Origin, Rubber-Sheet Pliability, and the C₆₀ as One Admissible Host Among Many · 2026 · DOIFurther development of the Yett Paradigm and its applications. Experimental verification of the Geodesic Pothole Mechanism. Study of the implications of the Yett Paradigm for the development of new propulsion systems.
The lack of a unified solution to the Millennium Prize Problems. The need for a new propulsion system that can access a propulsive force far exceeding any chemical or nuclear reaction.
The following study addresses the issue of calculating with precision the volume of a truncated right circular cone while provided with limited information on the dimensions of the object itself (being given the relation between the radii, the vertical heights, or the slanted heights).
Future research can focus on the applications of the experiment in biology and chemistry. Further study can be done on the combinatorial geometric blueprints based on Hamiltonian paths.
The paper identifies a gap in the understanding of the mathematics behind the morphogenesis of icosahedral shapes in nature. The study aims to fill this gap by presenting a novel mathematical video experiment.
Further research can be done to study properties of lattice polygons. Further research can be done to apply the results to other areas of geometry and computer science.
The paper identifies a gap in the study of convex lattice pentagons with three pairs of parallel sides and diagonals. The paper aims to characterize all obtained convex lattice pentagons of minimal area.
The question of extending the t-Feynman's configurations to tetrahedra appears naturally. There is a lack of understanding of the properties of t-Feynman's tetrahedra.
Heronian quadrilaterals have not been investigated thoroughly. The finiteness of amicable Heronian triangles and rectangles seems to end with this case.
Further study of the device's computational capabilities. Further analysis of the device's group-theoretic structure and physical encoding. Comparison with other ancient mathematical devices and instruments.
The Roman Dodecahedron as a Sexagesimal Computing Device: Group-Theoretic Structure, Physical Encoding, and Computational Signatures in Kepler and Ptolemy · 2026 · DOIThe functional identification of the Roman dodecahedron has resisted solution for centuries. The device's group-theoretic structure and physical encoding have not been previously understood. The device's computational capabilities have not been previously demonstrated.
The Roman Dodecahedron as a Sexagesimal Computing Device: Group-Theoretic Structure, Physical Encoding, and Computational Signatures in Kepler and Ptolemy · 2026 · DOITo study the properties of primitive Pythagorean triples using the result. To improve the efficiency of algorithms for generating Pythagorean triples using the result. To study the properties of other Diophantine equations using the result.
The Pell Spine as an Optimal Rational Separator: Extremal Resolution and Smooth Transport in Primitive Pythagorean Triples · 2026 · DOI
Most-cited papers in Mathematics and Applications
- Was Bowen Correct? The Relationship Between Differentiation and Triangulation · Contemporary Family Therapy · 2020 · 9 citations
- Response to David Morgan on Triangulation · Journal of Mixed Methods Research · 2022 · 6 citations
- On the Rational Values of Trigonometric Functions of Angles that Are Rational in Degrees · Mathematics Magazine · 2021 · 5 citations
- Gauss's Area Formula for Irregular Shapes · Ohio Journal of School Mathematics · 2024 · 5 citations
- Variations on the Calculi $$C_n$$ of da Costa · Studia Logica · 2025 · 4 citations
- Learning to measure the area of quadrilaterals through decomposition and recomposition · The Journal of Mathematical Behavior · 2022 · 4 citations
- The Centroid as a Nontrivial Area Bisecting Center of a Triangle · College Mathematics Journal · 2017 · 4 citations
- The Helical Periodic Table: The Pre-Mendeleev 3D Origin, Rubber-Sheet Pliability, and the C₆₀ as One Admissible Host Among Many · Zenodo (CERN European Organization for Nuclear Research) · 2026 · 3 citations
- Triangle angle sums related to translation curves in Sol geometry · Studia Universitatis Babes-Bolyai Matematica · 2022 · 3 citations
- Web Mercator Projection – One of Cylindrical Projections of an Ellipsoid to a Plane · Kartografija i geoinformacije · 2021 · 3 citations
Most recent work
- The Helical Periodic Table: The Pre-Mendeleev 3D Origin, Rubber-Sheet Pliability, and the C₆₀ as One Admissible Host Among Many · Zenodo (CERN European Organization for Nuclear Research) · 2026
- A Novel Summation Identity by Using Delannoy Triangle · Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics · 2026
- The Complete Solution of the Gasca-Maeztu Hypothesis · Open Access Journal of Artificial Intelligence and Technology · 2026
- A vanishing theorem in Siefring's intersection theory · International Journal of Mathematics · 2026
- From Thales and Pythagoras to companion rectangles · Journal of Mathematics and the Arts · 2026
- Geometry and Dress Groups with Non-symmetric Cost Functions · Vietnam Journal of Mathematics · 2026
- Multidimensional Generalized Geometric Progressions of Multiplicity One · International Journal of Mathematics And Computer Research · 2026
- Mahler equations for Zeckendorf numeration · International Journal of Algebra and Computation · 2026
- Minimum-Volume Primitive Equifacial Heronian Tetrahedron · Zenodo (CERN European Organization for Nuclear Research) · 2026
- CNRS-Tet: The Metric Correction from the Vierbein Formalism: Tetrad Re-derivation, Sub-leading Structure, and the Complex Geodesic Connection · Zenodo (CERN European Organization for Nuclear Research) · 2026
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