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Unsolved problem in science and mathematics
In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective
Open_problem
still an open problem for 4-dimensional manifolds. For differential manifolds, it is known to be false generally, but it is an open problem to determine
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
contain lists of notable problems. This includes unsolved problems, which may refer to several notable conjectures or open problems in various academic fields:
Lists_of_problems
number of important questions remain open in the area of physics beyond the Standard Model, such as the strong CP problem, determining the absolute mass of
List of unsolved problems in physics
List_of_unsolved_problems_in_physics
Unsolved problem in computer science
Unsolved problem in computer science If the solution to a problem can be verified in polynomial time, must the problem be solvable in polynomial time
P_versus_NP_problem
List of unsolved computational problems
NP? NC = P problem NP = co-NP problem P = BPP problem P = PSPACE problem L = NL problem PH = PSPACE problem L = P problem L = RL problem Unique games
List of unsolved problems in computer science
List_of_unsolved_problems_in_computer_science
Problem of determining if a Boolean formula could be made true
NP problem – one of the most important open problems in the theory of computing. Nevertheless, heuristic SAT-algorithms are able to solve problem instances
Boolean satisfiability problem
Boolean_satisfiability_problem
Seven mathematical problems with a US$1 million prize for each solution
artificial intelligence company OpenAI presented a proposed counterexample to the Navier–Stokes existence and smoothness problem, saying that they do not intend
Millennium_Prize_Problems
23 mathematical problems stated in 1900
Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. He intended to rival the master of French mathematics
Hilbert's_problems
On areas of shapes in 2d motion planning
to as the sofa constant. The exact value of the sofa constant is an open problem. The leading solution, by Joseph L. Gerver, has a value of approximately
Moving_sofa_problem
Book published in 2016
Open Problems in Mathematics is a book, edited by John Forbes Nash Jr. and Michael Th. Rassias, published in 2016 by Springer (ISBN 978-3-319-32160-8)
Open_Problems_in_Mathematics
Probability puzzle
The Monty Hall problem is a brain teaser, in the form of a probability puzzle, based nominally on the American television game show Let's Make a Deal
Monty_Hall_problem
Complexity class
problems are the hardest of the problems to which solutions can be verified quickly. NP is short for "nondeterministic polynomial-time" and a problem
NP-completeness
Problem in computer science
for the same problem. In its general form, it is equivalent to the hidden subgroup problem for the dihedral group. It is a major open problem to understand
Hidden_shift_problem
Conformance of AI to intended objectives
legitimacy. AI alignment can represent value pluralism. AI alignment is an open problem for modern AI systems and is a research field within AI. Aligning AI
AI_alignment
About polynomials in several variables
the two-dimensional problem and presented results that he had obtained in 1970–71. The conjecture, also called Jacobian problem, was subsequently widely
Jacobian_conjecture
Open problem on 3x+1 and x/2 functions
mathematics". However, though the Collatz conjecture itself remains open, efforts to solve the problem have led to new techniques and many partial results. Consider
Collatz_conjecture
Unanswered question in the study of consciousness
binding problem is the problem of how objects, background, and abstract or emotional features are combined into a single experience. The binding problem refers
Binding_problem
Unproven computational hardness assumption
examine this work. The k-SAT problem is a version of the Boolean satisfiability problem in which the input to the problem is a Boolean expression in conjunctive
Exponential_time_hypothesis
Quantum field theory
Unsolved problem in physics Yang–Mills theory and the mass gap. Quantum particles described by the theory have mass but the classical waves of the field
Yang–Mills_theory
Class in computational complexity theory
Unsolved problem in computer science N C = ? P {\displaystyle {\mathsf {NC}}{\overset {?}{=}}{\mathsf {P}}} More unsolved problems in computer science
NC_(complexity)
Mathematics problem
The 100 prisoners problem is a mathematical problem in probability theory and combinatorics. In this problem, 100 numbered prisoners must find their own
100_prisoners_problem
Unsolved problem in mathematics Which finite groups are BI-groups? More unsolved problems in mathematics Babai's problem is a problem in algebraic graph
Babai's_problem
Russian mathematician (born 1966)
proved the soul conjecture in Riemannian geometry, which had been an open problem for the previous 20 years. In 2002 and 2003, he developed new techniques
Grigori_Perelman
Millennium Prize Problem
specific statements of the problem as given by Charles Fefferman. On 8 September 2026, artificial intelligence company OpenAI claimed to have found a solution
Navier–Stokes existence and smoothness
Navier–Stokes_existence_and_smoothness
Conceptual issue in the philosophy of mind
actions. The problem divides into several distinct sub-problems, including the problem of causal exclusion, the problem of anomalism, and the problem of externalism
Problem_of_mental_causation
1979 conjecture in combinatorics
family? More unsolved problems in mathematics The union-closed sets conjecture, also known as Frankl’s conjecture, is an open problem in combinatorics posed
Union-closed_sets_conjecture
AI benchmark testbed for mathematical problem solving
Frontier Math Benchmark? Why Open Research Problems Expose True AI Reasoning". MindStudio. "FrontierMath: Open Problems - Unsolved Mathematical Challenges"
FrontierMath
McMullen problem is an open problem in discrete geometry named after Peter McMullen. In 1972, David G. Larman wrote about the following problem: Determine
McMullen_problem
Problem in theoretical computer science
accepts one of the two strings and rejects the other string. It is an open problem how large such an automaton must be, in the worst case, as a function
Separating_words_problem
Topics referred to by the same term
Open question may refer to: Open-ended question, a question that cannot be answered with a "yes" or "no" response Open problem, or open question, a known
Open_question
case n = 2, it is an open problem whether matrix mortality is decidable, but several special cases have been solved: the problem is decidable for sets
Matrix_mortality_problem
Unsolved problem about inscribing a square in a Jordan curve
Unsolved problem in mathematics Does every Jordan curve have an inscribed square? More unsolved problems in mathematics The inscribed square problem, also
Inscribed_square_problem
American mathematician and Nobel Laureate (1928–2015)
resolved Hilbert's nineteenth problem on regularity in the calculus of variations, which had been a well-known open problem for almost 60 years. In 1959
John_Forbes_Nash_Jr.
2026 scientific priority controversy
2026, artificial intelligence company OpenAI proposed a solution to the Navier–Stokes existence and smoothness problem with a smooth external force. However
Navier–Stokes priority controversy
Navier–Stokes_priority_controversy
expansion problem asks whether a greedy algorithm for finding Egyptian fractions with odd denominators always succeeds. It is an open problem. An Egyptian
Odd_greedy_expansion
Problem in discrete geometry
points are collinear and no four points are cocircular, then it is an open problem. Currently the best result is g gen ( n ) ≥ Ω ( n ) {\displaystyle
Erdős distinct distances problem
Erdős_distinct_distances_problem
Mathematical problem in operations research
In operations research, the cutting-stock problem is the problem of cutting standard-sized pieces of stock material, such as paper rolls or sheet metal
Cutting_stock_problem
Classification of algorithm
settle the P versus NP problem, considered the most important open problem in computer science and one of the Millennium Prize Problems. An example of a galactic
Galactic_algorithm
Optimization problem
Open-shop scheduling or open-shop scheduling problem (OSSP) is an optimization problem in computer science and operations research. It is a variant of
Open-shop_scheduling
Mathematical problem
Unsolved problem in mathematics How many colors are needed to color the plane so that no two points at unit distance are the same color? More unsolved
Hadwiger–Nelson_problem
in identifying problems have delayed statistics far more than difficulties in solving problems." A list of "one or two open problems" (in fact 22 of
List of unsolved problems in statistics
List_of_unsolved_problems_in_statistics
OEIS conjectures. In May 2026, OpenAI announced that an internal model had produced a disproof of the unit distance problem, and several mathematicians collaborated
List of mathematical discoveries made with artificial intelligence
List_of_mathematical_discoveries_made_with_artificial_intelligence
with many open problems. As in other areas of mathematics, such problems are often made public at professional conferences and meetings. Problems posed here
Problems_in_Latin_squares
Problem in computer science
complexity in the Turing machine model is open, as of 1997. The main difficulty is that, in order to solve the problem, the square-roots should be computed
Square-root_sum_problem
Probability of shared birthdays
In probability theory, the birthday problem asks for the probability that, in a set of n randomly chosen people, at least two will share the same birthday
Birthday_problem
18 mathematical problems stated in 1998
conjecture ... is one of the outstanding open problems in all of mathematics ... even the case n = 2 is still open. van den Essen, Arno (1997). "To Believe
Smale's_problems
Problem in number theory
problems in mathematics In the mathematics of sums of powers, it is an open problem to characterize the numbers that can be expressed as a sum of three cubes
Sums_of_three_cubes
On minimizing crossings in bicliques
Closing the gap between this lower bound and the upper bound remains an open problem. If edges are required to be drawn as straight line segments, rather
Turán's_brick_factory_problem
On dissections between polyhedra
The third of Hilbert's problems presented in 1900 was the first to be solved. The problem asks the following: Given any two polyhedra of equal volume,
Hilbert's_third_problem
Famous open problem in mathematics
It is an open problem in the field of number theory and is widely recognized as one of the most challenging unsolved mathematical problems. It is named
Birch and Swinnerton-Dyer conjecture
Birch_and_Swinnerton-Dyer_conjecture
Origami math problem
Svetlana Krat, Alexey S. Tarasov, and Ivan Yaschenko. One form of the problem remains open. There are several ways to define the notion of folding, giving different
Napkin_folding_problem
Unsolved problem in mathematics
Maass forms, the conjecture's counterpart for Maass forms is still an open problem, as the Deligne method which solves the holomorphic case does not work
Ramanujan–Petersson conjecture
Ramanujan–Petersson_conjecture
American artificial intelligence company
Prize Problem. OpenAI has filed for an initial public offering, but its size and timeline were not publicly disclosed. In December 2015, OpenAI was founded
OpenAI
1979 classic textbook on computational complexity theory
Spanning tree parity problem Partial order dimension Precedence constrained 3-processor scheduling This problem was still open as of 2016. Linear programming
Computers_and_Intractability
Mathematical problem involving optimal stopping theory
known as the marriage problem, the sultan's dowry problem, the fussy suitor problem, the googol game, and the best choice problem. Its solution is also
Secretary_problem
Conjecture in knot theory relating quantum invariants and hyperbolic geometry
branch of mathematics called knot theory, the volume conjecture is an open problem that relates quantum invariants of knots to the hyperbolic geometry of
Volume_conjecture
Hungarian mathematician (1913–1996)
necessarily appears. Overall, his work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics. He taught
Paul_Erdős
Mathematical problem
problem (also referred to as the Frobenius coin problem or Frobenius problem, after the mathematician Ferdinand Frobenius) is a mathematical problem that
Coin_problem
open problems. As in other areas of mathematics, such problems are often made public at professional conferences and meetings. Many of the problems posed
List of problems in loop theory and quasigroup theory
List_of_problems_in_loop_theory_and_quasigroup_theory
Unsolved problem in graph theory
More unsolved problems in mathematics In mathematics, the Oberwolfach problem is an open problem that may be formulated either as a problem of scheduling
Oberwolfach_problem
Informal fallacy
the proposition can be called unproven, undecided, inconclusive, an open problem or a conjecture. The term was likely coined by philosopher John Locke
Argument_from_ignorance
Two-dimensional packing problem
even for half-integer side lengths, remains an open problem. Square packing in a circle is a related problem of packing n unit squares into a circle with
Square_packing
Inherent difficulty of computational problems
computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem is a task
Computational complexity theory
Computational_complexity_theory
Algebraic structure
identities are somewhat more difficult (Kunen 1996). Phillips' problem is an open problem in the theory presented by J. D. Phillips at Loops '03 in Prague
Moufang_loop
Shape containing unit line segments in all directions
dimension n; it remains open for n>3. These questions belong to geometric measure theory. In 1920, while considering a problem of integration, Besicovitch
Kakeya_set
Points with no three in a line
intriguing open problems in additive combinatorics and Ramsey theory for over 20 years, highlighted, for instance, by blog posts on this problem from Fields
Cap_set
Computer software bug occurring in 2038
The year 2038 problem (also known as Y2038, Y2K38, Y2K38 superbug, or the Epochalypse) is a time computing problem that leaves some computer systems unable
Year_2038_problem
Topics referred to by the same term
Harmacy Open-end fund, collective investment which can issue and redeem shares at any time Open end (disambiguation) Open problem, or open question Open-question
Open-ended
Mathematical term in group theory
group has intermediate growth, thus providing an answer to an important open problem posed by John Milnor in 1968. The Grigorchuk group remains a key object
Grigorchuk_group
Topics referred to by the same term
equation for which a solution is sought Unknown, the ongoing answer to an open problem Unknown (video game) or Amnesia: The Dark Descent, a 2010 video game
Unknown
Hierarchical clustering of graph edges
treewidth for which the complexity on planar graphs is a well known open problem. The original algorithm for planar branchwidth, by Paul Seymour and Robin
Branch-decomposition
Hermite's problem is an open problem in mathematics posed by Charles Hermite in 1848. He asked for a way of expressing real numbers as sequences of natural
Hermite's_problem
This page lists notable open problems related to fair division - a field in the intersection of mathematics, computer science, political science and economics
List of unsolved problems in fair division
List_of_unsolved_problems_in_fair_division
Proposition in mathematics that is unproven
the single most important open problem in the field of computer science, it ranks as one of the seven Millennium Prize Problems selected by the Clay Mathematics
Conjecture
and output, but whether they can be solved in polynomial time is an open problem. A Boolean function takes as input an assignment of truth values to its
Monotone_dualization
2026 release of AI-generated mathematical solutions
intelligence company OpenAI released 722 manuscripts claiming to have solved a large number of open and unsolved major mathematical problems, via a GitHub repository
OpenAI bulk release of mathematical papers
OpenAI_bulk_release_of_mathematical_papers
Australian mathematician
in solving an important open problem that had been posed by Stephen Smale (the fourteenth problem on Smale's list of problems). He was an invited speaker
Warwick_Tucker
Rewriting system and type of formal grammar
rules, setting a new benchmark in L-system inference. There are many open problems involving studies of L-systems. For example: Characterisation of all
L-system
Open problem in computer science
unsolved problems in computer science In theoretical computer science and mathematics, the Hartmanis–Stearns conjecture is an open problem named after
Hartmanis–Stearns_conjecture
Extremal graph problem with bounded degree and diameter
Unsolved problem in mathematics Given positive integers d and k, what is the maximum possible number of vertices in a finite simple graph of maximum degree
Degree_diameter_problem
Classification of computer problems
Milind Sohoni. The goal of the program is to answer the most famous open problem in computer science – whether P = NP – by showing that the complexity
Geometric_complexity_theory
Concept in algebraic geometry
dimension at least 4 over fields of characteristic p, it is an open problem. Originally the problem of resolution of singularities was to find a nonsingular
Resolution_of_singularities
Class of problems solvable in polynomial time
class of problems decidable in polynomial space. PSPACE is equivalent to NPSPACE by Savitch's theorem. Again, whether P = PSPACE is an open problem. To summarize:
P_(complexity)
Solution of some Diophantine equation
substituted for quantification over the integers is a notoriously hard open problem. The MRDP theorem (so named for the initials of the four principal contributors
Diophantine_set
equivalence relation. While this is still an open problem, some partial results are known. Another open problem in the area is whether a countable increasing
Hyperfinite equivalence relation
Hyperfinite_equivalence_relation
Method to solve optimization problems
algorithm? More unsolved problems in computer science There are several open problems in the theory of linear programming, the solution of which would represent
Linear_programming
Open problem in natural language processing
computational linguistics, word-sense induction (WSI) or discrimination is an open problem of natural language processing, which concerns the automatic identification
Word-sense_induction
Graphs formed by a hypercube's edges and vertices
question whether every matching extends to a Hamiltonian cycle remains an open problem. The hypercube graph Q n {\displaystyle Q_{n}} (for n > 1 {\displaystyle
Hypercube_graph
Austrian-Canadian mathematician
as Moser's worm problem and as of 2026 remains an open problem. Bell number Berlekamp switching game Salem–Spencer set Secretary problem Tournament (graph
Leo_Moser
Computational graph problem
science and graph theory, the Canadian traveller problem (CTP) is a generalization of the shortest path problem to graphs that are partially observable. In
Canadian_traveller_problem
Subfield of computational complexity theory
establishing that SAT requires more than linear time is open. Meanwhile, many fundamental problems have textbook algorithms whose running times haven't been
Fine_grained_complexity
Concept in theoretical computer science
used to systematically solve many open problems in mathematics. However, current results on the busy beaver problem suggest that this will not be practical
Busy_beaver
Planar maps require at most four colors
Hedetniemi, Stephen T. (eds.), Graph Theory: Favorite Conjectures and Open Problems, II, Problem Books in Mathematics, Springer International Publishing, pp. 115–133
Four_color_theorem
Complexity class
hashing. PPP contains PPAD as a subclass (strict containment is an open problem). This is because End-of-the-Line, which defines PPAD, admits a straightforward
PPP_(complexity)
Problem in number theory on equal totients
and in 1922, he retracted his claim and stated the conjecture as an open problem. The totient function φ ( n ) {\displaystyle \varphi (n)} is equal to
Carmichael's totient function conjecture
Carmichael's_totient_function_conjecture
Selected Unsolved Problems in Coding Theory. New York: Springer. Longo, Giuseppe (1975). Information theory: new trends and open problems. Springer. ISBN 9783211813782
List of unsolved problems in information theory
List_of_unsolved_problems_in_information_theory
problem 90, the planar unit distance problem, was solved by an internal OpenAI model in May 2026. Erdős problem 522 on random Littlewood polynomials was
List of conjectures by Paul Erdős
List_of_conjectures_by_Paul_Erdős
Concept in graph theory
Comb. Theory Ser B, 30 (1981), 130–135. [1], Open Problem Garden. [2], Open Problem Garden. [3], Open Problem Garden. Zhang, Cun-Quan (1997). Integer Flows
Nowhere-zero_flow
Computational geometry problem
also known, but the case d ≥ 3 remains an open problem. In 1977, Victor Klee considered the following problem: given a collection of n intervals in the
Klee's_measure_problem
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