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Open problem

In mathematics, an open problem is an unsolved question, conjecture, or challenge that lacks a complete proof, disproof, or resolution, despite significant efforts by researchers.^[1] These problems arise across all branches of mathematics, from number theory and geometry to topology and computational complexity, and they represent fundamental gaps in current understanding.^[2] Open problems serve as vital catalysts for mathematical progress, guiding research directions, fostering interdisciplinary connections, and motivating generations of mathematicians to explore uncharted territories.^[3] Historically, landmark lists such as David Hilbert's 23 problems presented at the 1900 International Congress of Mathematicians have profoundly influenced the development of modern mathematics, with many spurring entire subfields like functional analysis and quantum mechanics.^[4] Similarly, the Clay Mathematics Institute's Millennium Prize Problems, announced in 2000, highlight seven particularly profound open challenges—such as the Riemann Hypothesis on prime distribution and the P versus NP question in computational complexity—each carrying a $1 million prize to incentivize solutions and underscore their enduring significance.^[2] While some open problems remain accessible to amateurs or students, others demand advanced techniques and have resisted resolution for centuries, as exemplified by the quest for odd perfect numbers, dating back to ancient times.^[5] Solving an open problem often yields not only the direct answer but also transformative tools and insights applicable far beyond the original question, reinforcing the dynamic and exploratory nature of mathematical inquiry.^[2]

Definition and Characteristics

Core Definition

An open problem in mathematics is a well-posed question or conjecture that has not been resolved despite substantial efforts by researchers. A problem is considered well-posed if it features a clear, precise statement and allows for verifiable resolution upon solution, ensuring that any purported answer can be objectively checked for correctness. This distinguishes open problems from vaguely formulated queries, emphasizing their suitability for rigorous investigation.^[1] Open problems encompass a broader category than conjectures alone; while conjectures are specific unproven statements typically suspected to be true based on evidence or intuition, open problems include both such conjectures and neutral unsolved questions lacking a strong presumed outcome.^[6] They are commonly phrased to invite proof, disproof, or computation, such as "Prove or disprove that every even integer greater than 2 is the sum of two primes" or "Find a closed-form expression for the Riemann zeta function at s=3." The term "open problem" emerged in mathematical literature toward the end of the 19th century, reflecting the growing recognition of persistent unsolved challenges.^[7] This usage gained prominence with David Hilbert's 1900 address, which presented 23 such problems as key directions for future research.^[8]

Key Characteristics

Open problems in mathematics exhibit remarkable persistence, frequently enduring for decades or centuries without resolution despite intensive investigation by experts worldwide. This longevity stems from the inherent complexity of the questions posed, which resist straightforward approaches and require innovative breakthroughs. For example, the Riemann Hypothesis, articulated in 1859, has remained unsolved for 166 years as of 2025, highlighting how such problems can define research agendas across generations.^[9] A core attribute is verifiability, demanding that any resolution—whether affirmative proof or disproof—be rigorously established through logical deduction and subjected to communal scrutiny for objectivity and correctness. Solutions must not merely appear plausible but achieve consensus via publication in esteemed peer-reviewed journals, followed by sustained acceptance over time. The Clay Mathematics Institute's criteria for its Millennium Prize Problems exemplify this, stipulating that a submission gains the prize only after two years of general endorsement by the mathematical community, ensuring enduring validity.^[10] Partial results, such as solutions to special cases or empirical confirmations, do not suffice for closure, as they leave the general formulation intact. These problems often evolve dynamically through incremental advances that illuminate aspects without fully resolving them, thereby sustaining interest and refining the path forward. Approximations, counterexamples in restricted scenarios, or strengthened bounds contribute to this progression; for instance, in the Riemann Hypothesis, partial verifications include Hardy's 1914 demonstration of infinitely many zeros on the critical line and computational checks confirming the hypothesis for the first 10^{32} zeros as of the 2020s, yet the universal truth persists as open. This layered development distinguishes open problems from settled ones, fostering ongoing refinement. In terms of difficulty, open problems span a spectrum from major challenges that reshape entire disciplines to minor inquiries confined to narrow domains. Major problems, like those in Stephen Smale's 1998 list of 18 pivotal unsolved issues for the 21st century—including the Riemann Hypothesis—carry sweeping implications for theoretical foundations and applications.^[11] Minor problems, by contrast, involve targeted gaps in specialized theories, resolved more readily but still requiring precise innovation within their scope. The Riemann Hypothesis exemplifies a major open problem, embodying persistence, verifiability demands, evolutionary progress, and profound influence.

Historical Context

Early Historical Examples

One of the earliest recognized open problems in mathematics arose in ancient Greece around the 5th century BCE, known as squaring the circle, which challenged mathematicians to construct a square with the same area as a given circle using only a compass and straightedge.^[12] This problem, first attempted by figures like Anaxagoras of Clazomenae during his imprisonment and later advanced by Hippocrates of Chios through his quadrature of lunes, remained unsolved for over two millennia, captivating scholars across cultures including in India, China, and the Arab world.^[12] Although Hippocrates achieved partial successes by squaring certain crescent-shaped regions, the full circle eluded resolution until the late 19th century, highlighting the enduring puzzle of transcendental numbers like π.^[12] During the same era, another classical challenge emerged: trisecting an arbitrary angle using compass and straightedge, formulated around 430 BCE by Hippias of Elis as part of broader efforts in geometric construction.^[13] This problem, alongside squaring the circle and doubling the cube, spurred innovations such as the quadratrix curve invented by Hippias and conchoid curves by Nicomedes in the 3rd century BCE, yet it persisted as open through the medieval and Renaissance periods despite mechanical approximations by Archimedes and conic-based approaches by Apollonius of Perga.^[13] The quest for trisection influenced the study of conic sections and highlighted the limitations of Euclidean tools, remaining unresolved until Pierre Wantzel's 1837 proof of its impossibility for general angles.^[13] In the 17th century, Pierre de Fermat posed what would become one of the most famous open problems in number theory: his Last Theorem, stated in 1637 in the margin of a copy of Diophantus's Arithmetica, claiming no positive integers a, b, c satisfy a^n + b^n = c^n for n > 2. Partial progress followed, with Leonhard Euler providing a proof for n=3 in 1770 using infinite descent, though initially flawed and later corrected, and further cases like n=5 by Adrien-Marie Legendre and Peter Gustav Lejeune Dirichlet in 1825, n=7 by Gabriel Lamé in 1839, and n=14 by Dirichlet in 1832. By the mid-19th century, Ernst Kummer extended results to "regular" primes using his theory of ideal numbers, yet the general case endured as open, driving advancements in algebraic integers. These pre-20th century problems, though not labeled as "open problems" in modern terms, profoundly shaped geometry and number theory by necessitating new tools and theories, from conic sections in antiquity to ideal factorization in the 19th century, laying groundwork for later mathematical formalization.^[12]^[13]

Development in Modern Mathematics

In 1900, David Hilbert presented 23 foundational problems at the Second International Congress of Mathematicians in Paris, outlining key challenges intended to guide mathematical research for the ensuing century. These problems spanned diverse areas including number theory, algebra, and geometry, and their influence extended far beyond individual solutions, shaping entire subfields of mathematics. Assessments indicate that 19 of these problems have been fully resolved, one has been partially addressed, and three remain open, demonstrating the enduring impact of Hilbert's vision.^[14]^[15] Building on this tradition, later collections further formalized the role of open problems in directing research agendas. In 1998, Steve Smale proposed 18 problems, primarily in dynamical systems, numerical analysis, and theoretical computer science, as priorities for the 21st century; these were published in The Mathematical Intelligencer at the request of the International Mathematical Union. Two years later, in 2000, the Clay Mathematics Institute announced the Millennium Prize Problems—a set of seven profound challenges in fields like topology, fluid dynamics, and complexity theory—each offering a $1 million prize to incentivize breakthroughs.^[16]^[2] Major conferences have played a pivotal institutional role in highlighting open problems, with the International Congress of Mathematicians featuring discussions on unsolved questions in various sessions to promote collaborative exploration. In the digital age, tools like the Open Problem Garden, a wiki-style database launched in 2007, have enabled global cataloging and real-time collaboration on open problems, allowing mathematicians to contribute descriptions, updates, and references across specialties.^[17]^[18]

Importance and Impact

Role in Advancing Research

Open problems serve as powerful catalysts for the development of new mathematical techniques and methodologies, often pushing researchers to invent tools that extend far beyond the original challenge. For instance, efforts to resolve Fermat's Last Theorem in the 19th century motivated Ernst Kummer to introduce the concept of ideal numbers, which addressed the failure of unique factorization in certain cyclotomic fields and laid foundational groundwork for modern algebraic number theory.^[19] Similarly, the full proof by Andrew Wiles in 1994 relied on advanced connections between elliptic curves and modular forms, spurring innovations in algebraic geometry and Galois representations that continue to influence contemporary number theory research.^[20] These examples illustrate how persistent open problems drive the creation of novel frameworks, transforming isolated conjectures into engines for broader methodological progress. The interdisciplinary spillover from mathematical open problems further amplifies their role in advancing research, as solutions or partial advances often migrate to other fields, reshaping scientific inquiry. A prominent case is the Navier-Stokes equations, whose unresolved existence and smoothness in three dimensions not only challenge pure mathematicians but also profoundly impact physics and engineering by governing fluid motion in applications from aerodynamics to weather modeling.^[21] The equations' study has led to hybrid approaches combining partial differential equations with physical simulations, enabling breakthroughs in turbulence modeling that bridge theoretical mathematics and experimental physics.^[22] Such cross-pollination encourages mathematicians to collaborate with physicists, fostering hybrid techniques that address real-world phenomena while deepening abstract understanding. Open problems also build vibrant research communities by inspiring dedicated collaborations, workshops, and specialized publications that sustain long-term progress. Institutions like the American Institute of Mathematics organize workshops explicitly around open problems, convening experts to explore challenges and develop collaborative strategies that yield incremental advances.^[23] Similarly, programs such as the Institute for Advanced Study's PCMI workshops facilitate group work on unresolved questions, promoting diverse participant engagement and new partnerships across subfields.^[24] These efforts extend to journals and proceedings that report partial results, creating a feedback loop where community input refines approaches and accelerates collective momentum. Advancing or solving an open problem acts as a key metric of research progress, elevating individual careers while potentially shifting entire paradigms in mathematics. Landmark resolutions, such as those inspired by historical collections like Hilbert's 1900 problems, have historically redirected research agendas and earned recipients prestigious recognition, underscoring their role in career-defining achievements.^[8] For researchers, partial progress on such challenges demonstrates methodological ingenuity, often leading to paradigm shifts—like the unification of disparate theories in number theory following Fermat's resolution—that redefine field boundaries and inspire future generations.^[25]

Incentives and Recognition

Solving open problems in mathematics often comes with substantial monetary incentives designed to spur innovation and collaboration. The most prominent example is the Clay Mathematics Institute's Millennium Prize Problems, announced in 2000, which offer a $1 million award for a correct solution to each of seven longstanding challenges in various fields of mathematics.^[2] To date, only one has been solved: the Poincaré conjecture, proved by Grigori Perelman in 2002–2003, though he declined the prize in 2010. Beyond monetary prizes, prestigious awards like the Fields Medal recognize significant progress or solutions to open problems, particularly for mathematicians under 40. In 2006, Perelman was awarded the Fields Medal by the International Mathematical Union for his proof of the Poincaré conjecture, highlighting its revolutionary impact on geometry and geometric analysis, though he also declined this honor.^[26] Similarly, in 2022, Maryna Viazovska received the Fields Medal for her proof that the E8 lattice achieves the densest packing of spheres in eight dimensions, resolving a century-old problem in discrete geometry.^[27] For older mathematicians, the Abel Prize serves a comparable role; Andrew Wiles was awarded it in 2016 for his 1994 proof of Fermat's Last Theorem, a problem that had eluded solutions for over 350 years.^[28] Non-monetary recognition further incentivizes efforts on open problems through enduring prestige and academic priority. Solvers gain lasting fame, often becoming synonymous with their achievement—such as Wiles with Fermat's Last Theorem—leading to high-profile publications, invitations to speak at major conferences, and the establishment of their name in historical contexts.^[28] The psychological allure of cracking a long-standing puzzle provides personal motivation, fostering a sense of communal triumph within the mathematical community, where even unsuccessful attempts can yield breakthroughs; for instance, prolonged efforts on the four-color theorem spurred the development of algebraic graph theory as a new subfield.^[29]

Notable Examples by Field

In Pure Mathematics

One of the most famous open problems in pure mathematics is the Riemann Hypothesis, formulated by Bernhard Riemann in 1859. It states that all non-trivial zeros of the Riemann zeta function \zeta(s), defined for complex numbers s with real part greater than 1 as \zeta(s) = \sum_{n=1}^\infty n^{-s} and extended analytically to the complex plane, lie on the critical line where the real part of s is $1/2. This hypothesis has profound implications for the distribution of prime numbers, as it would provide the optimal error term in the Prime Number Theorem, which approximates the number of primes up to x as \pi(x) \sim x / \log x. Partial progress includes Atle Selberg's 1942 proof that a positive proportion of the non-trivial zeros lie on the critical line, with subsequent improvements raising the lower bound to at least 41.729% as of 2025.^[30] Extensive computational verification confirms that the first $10^{13} non-trivial zeros all lie on this line, with computations extending to heights around $10^{32} as of recent efforts. As of 2025, no counterexamples have been found, and theoretical approaches, such as connections to random matrix theory, continue to support the hypothesis without a full proof. Another prominent unsolved problem is P versus NP, posed by Stephen Cook in 1971. It asks whether every problem whose solution can be verified in polynomial time (NP) can also be solved in polynomial time (P), formally: does P = NP?^[31] This question arises in computational complexity theory and impacts fields like optimization and cryptography, as resolving it in favor of P ≠ NP would confirm inherent computational hardness for certain problems. Despite decades of research, there has been no progress toward a resolution, with barriers such as relativization, natural proofs, and algebrization showing that common proof techniques fail to separate the classes.^[32] As of 2025, the problem remains open, with ongoing workshops exploring lower bounds and fine-grained complexity, but no breakthroughs altering its status.^[33] The Birch and Swinnerton-Dyer Conjecture, proposed by Bryan Birch and Peter Swinnerton-Dyer in the 1960s based on early computational experiments, relates the arithmetic of elliptic curves to their analytic properties. For an elliptic curve E over the rationals, it asserts that the rank of the Mordell-Weil group E(\mathbb{Q})—the dimension of the group of rational points—is equal to the order of vanishing of the L-function L(E, s) at s=1, and further predicts the precise leading term involving the Sha group and regulators.^[34] Partial results confirm the conjecture for ranks 0 and 1: for rank 0, the L-function does not vanish at s=1 if and only if E(\mathbb{Q}) is finite, proven via descent methods; for rank 1, the Gross-Zagier formula and Kolyvagin's Euler systems establish the equality and non-vanishing of the derivative. As of 2025, the conjecture holds for these low-rank cases across many families of curves, with average rank bounds supporting low ranks, but remains unproven for ranks 2 and higher, where Iwasawa theory and p-adic methods provide ongoing theoretical approaches.

In Applied Fields

In applied mathematics and physics, the Navier-Stokes existence and smoothness problem stands as a cornerstone open challenge, originating from the equations formulated by Claude-Louis Navier and George Gabriel Stokes in the 1840s to describe the motion of viscous fluids. The core question is whether, for any smooth initial velocity field in three-dimensional space, there exists a smooth solution to the incompressible Navier-Stokes equations that remains smooth for all positive times, or if singularities—points of infinite velocity—can develop in finite time.^[35] This Millennium Prize Problem, offering a $1 million award from the Clay Mathematics Institute, has profound implications for modeling real-world phenomena like turbulence in aerodynamics, ocean currents, and weather prediction, where breakdowns in smoothness could explain chaotic fluid behaviors but remain unproven.^[21] As of 2025, numerical simulations have advanced understanding through AI-driven discoveries of unstable singularities in related fluid dynamics equations, such as those reported by Google DeepMind, which suggest potential blow-up scenarios but do not resolve the global existence question.^[36] Another prominent open problem in applied number theory is the Collatz conjecture, proposed by Lothar Collatz in 1937, which posits that for any positive integer n, repeatedly applying the rule—if n is even, divide by 2; if odd, replace with 3n + 1—will eventually reach 1. Despite its deceptive simplicity, no general proof exists, though the conjecture has been computationally verified for all starting values up to approximately 2⁷¹ (about 2.36 × 10²¹) as of early 2025, using optimized algorithms on high-performance computing clusters.^[37] This iterative process finds applications in dynamical systems analysis and computational testing of recursive algorithms, highlighting patterns in integer sequences that inform software verification and chaos theory in engineering contexts. Recent progress includes refined verification techniques that extend the checked range by leveraging parallel processing and heuristic optimizations, yet the conjecture's universality for all integers eludes analytical closure.^[38] In computer science and quantum information theory, scalability challenges in quantum computing, particularly for implementing Shor's algorithm, represent a critical open frontier with direct implications for cryptography and optimization. Introduced by Peter Shor in 1994, the algorithm promises to factor large integers exponentially faster than classical methods, potentially rendering widely used public-key encryption like RSA obsolete if executed on a sufficiently large, fault-tolerant quantum computer. However, realizing this requires overcoming error correction hurdles, as quantum bits (qubits) are highly susceptible to decoherence and noise, necessitating thousands of physical qubits per logical qubit to suppress errors below thresholds for reliable computation—a scalability issue unresolved as of 2025.^[39] Breakthroughs in 2025, such as Quantinuum's collaboration with Princeton and NIST demonstrating improved logical qubit fidelity through advanced surface code implementations, and Microsoft's development of four-dimensional error-correcting codes, have reduced error rates in small-scale systems but fall short of the million-qubit scale needed for practical Shor's execution on cryptographically relevant numbers.^[40]^[41] These advances underscore the real-world stakes, driving investments in hybrid quantum-classical simulations for drug discovery and materials science while highlighting the ongoing quest for viable approximations in noisy intermediate-scale quantum devices.

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