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Conjecture

In mathematics, a conjecture is a proposition or statement that is proposed as true based on empirical evidence, patterns, or incomplete proofs, but which lacks a rigorous formal proof or disproof.^[1] These unverified assertions serve as foundational hypotheses that inspire further investigation and often drive significant advancements in mathematical research.^[2] Unlike theorems, which are established truths, conjectures remain open questions until resolved, and they can be either proven correct, refuted by counterexamples, or persist indefinitely as unsolved problems.^[3] Conjectures have played a pivotal role in the development of mathematics since ancient times, with early examples appearing in Greek geometry and number theory, though the modern concept solidified during the Enlightenment era.^[4] One of the most enduring is the Goldbach conjecture, proposed in 1742 by Christian Goldbach, which states that every even integer greater than 2 can be expressed as the sum of two prime numbers; it has been verified computationally for numbers up to extremely large values but remains unproven.^[4] Similarly, the Riemann hypothesis, formulated by Bernhard Riemann in 1859, posits that all non-trivial zeros of the Riemann zeta function have a real part of 1/2, with profound implications for the distribution of prime numbers; it is considered one of the most important unsolved problems in mathematics.^[5] Many conjectures have been resolved over time, transforming into theorems that reshape fields like topology and algebra. For instance, the Poincaré conjecture, stated by Henri Poincaré in 1904, asserted that every simply connected, closed 3-manifold is topologically equivalent to the 3-sphere; it was proven by Grigori Perelman in 2003 using Ricci flow techniques, earning him the Fields Medal (which he declined).^[6] Other famous cases include Fermat's Last Theorem, conjectured in 1637 and proven by Andrew Wiles in 1994, which states that no three positive integers a, b, and c satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2.^[7] Conversely, some conjectures, like the Euler conjecture on sums of powers (disproven in 1966), highlight the risk of counterexamples emerging after centuries of apparent validity.^[3] The significance of conjectures extends to contemporary mathematics, where six unsolved ones form the Millennium Prize Problems, established by the Clay Mathematics Institute in 2000, each carrying a $1 million prize for a correct solution.^[8] These include the Birch and Swinnerton-Dyer conjecture on elliptic curves, the Hodge conjecture in algebraic geometry, and the P versus NP problem in computational complexity, underscoring how conjectures continue to challenge and unify diverse mathematical disciplines.^[9]^[10] Through computational verification, heuristic arguments, and interdisciplinary approaches, mathematicians persist in testing and refining these ideas, often leading to breakthroughs in unrelated areas.^[11]

Fundamentals

Definition

In mathematics, a conjecture is a proposition that is consistent with known data but has neither been rigorously verified nor shown to be false.^[12] It represents a mathematical statement proposed as true based on incomplete evidence, such as patterns observed in limited cases, yet lacking a formal proof or disproof.^[13] This unproven status distinguishes conjectures from other foundational elements in logic and mathematics, positioning them as tentative assertions that invite further investigation. Key characteristics of conjectures include their empirical origins and provisional nature; they often emerge from recognizing recurring patterns across examples, but they remain open to challenge until resolved.^[13] In contrast, axioms are propositions regarded as self-evidently true without requiring proof, serving as unassailable starting points for deductive reasoning.^[14] Theorems, meanwhile, are statements that have been demonstrated to be true through accepted mathematical operations, arguments, and prior established results.^[15] Unlike hypotheses in scientific contexts, which may be testable through experimentation, mathematical conjectures rely on logical deduction for validation. Conjectures play a vital logical role in mathematical research by acting as catalysts for exploration, directing efforts toward proofs, counterexamples, or deeper theoretical developments.^[16] For instance, the Goldbach conjecture posits that every even integer greater than 2 can be expressed as the sum of two prime numbers, a simple arithmetic claim derived from initial verifications that has spurred extensive number theory investigations.^[17]

Historical Origins

The term "conjecture" derives from the Latin coniectūra, meaning "a putting together" or "interpretation," rooted in conicere ("to throw together" or "to infer"), and entered English in the late 14th century via Old French, initially denoting a guess or inference based on incomplete evidence.^[18] In ancient mathematics, conjectures emerged as observational insights that guided early proofs, particularly among the Pythagoreans around 500 BCE, who proposed relationships such as the sum of the first n consecutive odd numbers equaling n² (e.g., 1 + 3 + 5 = 9), viewing numbers as embodying mystical properties and using these patterns to explore arithmetic and geometry.^[19] Euclid's Elements (circa 300 BCE) formalized many such ideas through rigorous proofs, transforming prior unproven assertions into theorems; for instance, his demonstration of the infinitude of primes in Book IX, Proposition 20, built on implied earlier speculations about prime distribution, while leaving some geometric assumptions, such as the parallel postulate, as unproven postulates that were later challenged, leading to the development of non-Euclidean geometries.^[20] During the medieval and Renaissance periods, interest in Diophantine problems—equations seeking integer solutions—was revived, initially systematized by Diophantus of Alexandria in the 3rd century CE and expanded by Islamic scholars like al-Karaji (circa 1000 CE), who worked on algebraic identities and indeterminate equations.^[21] This tradition influenced European mathematicians, culminating in Pierre de Fermat's 1637 marginal note in Diophantus's Arithmetica, where he conjectured that no positive integers a, b, c satisfy aⁿ + bⁿ = cⁿ for n > 2, claiming a proof that remained unpublished and became known as Fermat's Last Theorem, exemplifying proto-conjectures as provocative challenges without full justification.^[22] The 19th century saw conjectures evolve into formal hypotheses within burgeoning fields like number theory, with Bernhard Riemann's 1859 paper "On the Number of Primes Less Than a Given Magnitude" proposing that all non-trivial zeros of the Riemann zeta function have real part 1/2, linking prime distribution to complex analysis and establishing a model for precise, research-driving statements.^[23] By 1900, David Hilbert's 23 problems, presented at the International Congress of Mathematicians, elevated conjectures to structured targets for collective inquiry, including the continuum hypothesis (Problem 1) and the Riemann hypothesis (Problem 8), emphasizing their role in testing theories, fostering methodological advances, and unifying mathematical progress from informal ancient guesses to modern axiomatic pursuits.^[24]

Formulation and Types

Empirical Foundations

Conjectures in mathematics often originate through inductive reasoning, where mathematicians observe patterns in specific instances and generalize them to broader statements without a deductive proof. This process involves verifying the proposed relation for a finite number of cases, such as checking small positive integers to identify recurring behaviors, to form a tentative assertion that appears plausible. Unlike deductive reasoning, which guarantees truth from axioms, induction provides only suggestive evidence, as the pattern may fail for unexamined cases.^[25] The role of computation has evolved significantly in building empirical support for conjectures, transitioning from manual verifications in early mathematics to extensive automated checks today. For instance, early efforts relied on hand calculations for small values, but modern supercomputers enable testing up to extraordinarily large bounds, such as verifying Goldbach's conjecture—that every even integer greater than 2 is the sum of two primes—for all even numbers up to 4 × 10^18 as of 2014.^[26] Similarly, for the twin prime conjecture, which posits infinitely many pairs of primes differing by 2, computational searches have identified twin primes up to numbers exceeding 10^18, with the largest known pair having 388,342 digits as of 2016; exhaustive counts of twin prime pairs are known up to 10^18.^[27]^[28] These efforts demonstrate the conjecture's resilience but remain inductive, as they cannot confirm infinitude. While empirical foundations lend plausibility to conjectures, they carry inherent limitations, as counterexamples may lurk beyond tested ranges, underscoring their tentative status. The twin prime pattern, observed in small primes like (3,5), (5,7), and (11,13), builds empirical confidence through repeated occurrences, yet probabilistic models suggest the density of such pairs diminishes asymptotically, potentially allowing eventual scarcity without disproving infinitude. Heuristic evidence, including these probabilistic frameworks like the Hardy-Littlewood conjecture, further bolsters support by estimating expected frequencies based on prime distributions, though they do not constitute proof.^[29] Mathematical conjectures differ from scientific hypotheses in their specificity and testing methods; conjectures propose precise, universal statements about mathematical objects, evaluated through logical or computational means, whereas hypotheses in science are broader explanations testable via empirical experiments and potentially falsifiable by observation. This distinction highlights conjectures' reliance on pattern recognition within abstract structures rather than physical data.^[30]

Formal Statements

In mathematical logic, conjectures are formalized using the precise syntax of first-order predicate logic, which allows for the expression of statements involving variables, predicates, functions, quantifiers, and logical connectives. A common structure employs the universal quantifier ∀ to assert properties holding for all elements in a domain, combined with predicates that define specific conditions. For instance, a conjecture might take the form ∀x ∈ D, P(x), where D is the domain (such as the natural numbers ℕ) and P(x) is a predicate expressing a property of x. This formalization ensures that conjectures are unambiguous and amenable to rigorous analysis within axiomatic systems. Predicates can involve arithmetic relations, set memberships, or other mathematical concepts, while existential quantifiers ∃ may appear in subformulas to claim the existence of objects satisfying certain criteria. Logical connectives like implication (→), conjunction (∧), and negation (¬) link these components to build complex statements. For example, the conjecture that there are infinitely many primes p such that p-1 is square-free can be expressed as: there exist infinitely many prime numbers p for which p-1 has no squared prime factors, or more symbolically, the set {p ∈ ℙ | μ(p-1) ≠ 0} is infinite, where ℙ denotes the primes and μ is the Möbius function (with μ(k) = 0 if k is not square-free). Within mathematical literature, formal conjectures are frequently presented as open problems or auxiliary statements in research papers, serving as unproven assumptions that underpin further results or highlight directions for investigation. They may appear as proposed lemmas whose validity is suspected but not yet established, enabling authors to derive conditional theorems under the conjecture's assumption. For instance, a paper might state a conjecture explicitly before exploring its implications for related theorems, thereby integrating it into the broader discourse without claiming a proof.^[31] Conjectures form a significant subset of unsolved problems in mathematics, where their formal statements allow for partial progress through theorems that hold assuming the conjecture's truth. Such conditional results often reveal the conjecture's far-reaching implications, motivating efforts toward resolution while providing tools for applications in adjacent fields.^[32] In the context of formal axiomatic systems like Zermelo-Fraenkel set theory with the axiom of choice (ZFC), conjectures are encoded as first-order sentences in the language of set theory, which consists of variables, the membership relation ∈, logical connectives, and quantifiers over sets. These sentences may be provable from the ZFC axioms, refutable, or independent, meaning neither provable nor refutable within the system.^[32] Gödel's incompleteness theorems, published in 1931, demonstrate that in any consistent formal system capable of expressing basic arithmetic (such as ZFC), there exist true statements that cannot be proved or disproved within the system, implying that some conjectures formulated as such sentences may be inherently independent of the axioms.^[33]

Resolution Approaches

Proofs and Verification

Proving a conjecture involves establishing its truth through rigorous logical deduction from accepted axioms and previously proven theorems, transforming it from a hypothesized statement into a verified theorem. Common methods include direct proof, where the conjecture's formal statement is assumed and logically derived step-by-step using definitions, axioms, and inference rules to reach the desired conclusion; proof by contradiction, which assumes the negation of the conjecture and demonstrates that this leads to an logical impossibility; and mathematical induction, particularly useful for statements involving natural numbers, where a base case is proven and the inductive step shows that if the statement holds for some k, it holds for k+1.^[34]^[35]^[36] Often, proving a conjecture requires the development of new theorems, lemmas, or mathematical tools as auxiliary results to bridge gaps in existing knowledge, providing the necessary framework for the main argument. For example, the role of innovative concepts like modular forms has been pivotal in resolving longstanding conjectures by enabling novel connections between disparate areas of number theory.^[37] These auxiliary constructions not only support the primary proof but also frequently open avenues for further mathematical exploration. Once a proof is constructed, verification ensures its correctness through peer review by experts in the field, who scrutinize the logical steps, assumptions, and derivations for errors or gaps, typically via submission to reputable mathematical journals. In complex cases, especially post-2000, computer-assisted verification has become increasingly prevalent, employing proof assistants like the Coq system to formally check every inference in a mechanized environment, reducing human oversight risks and handling exhaustive case analyses infeasible manually.^[38]^[39]^[40] The successful proof of a conjecture elevates it to the status of a theorem, definitively resolving an open question and often catalyzing new research directions by revealing deeper structural insights or applications in related fields. This elevation not only solidifies foundational knowledge but also inspires subsequent conjectures and proofs, contributing to the progressive architecture of mathematics.^[16]

Disproofs and Counterexamples

Disproofs of conjectures in mathematics occur when a proposed statement is shown to be false, most commonly through the identification of a counterexample—a single instance that satisfies the conjecture's premises but violates its conclusion. This method is definitive because, for universal statements claiming a property holds for all cases, one violation suffices to refute the entire claim.^[41] Constructing counterexamples often involves exhaustive searches over finite domains or the strategic selection of values that exploit potential weaknesses in the conjecture's structure. Algebraic manipulations, such as rearranging terms or applying modular arithmetic, can reveal inconsistencies without broad enumeration, while computational approaches enable exploration of vast parameter spaces that manual methods cannot handle. In the 20th century, the advent of digital computers marked a significant shift toward automated disproofs, allowing researchers to systematically test hypotheses at scales previously unimaginable and accelerating the refutation of longstanding ideas.^[42]^[41] A prominent example is the 1966 disproof of Euler's sum of powers conjecture, which posited that at least k positive kth powers are needed to sum to another kth power for k > 2. Lander and Parkin used a direct computer search to find the counterexample $27^5 + 84^5 + 110^5 + 133^5 = 144^5, requiring only four terms instead of five. The consequences of such disproofs extend beyond mere refutation; they narrow the scope of the original problem, often prompting the formulation of revised, "weakened" conjectures that hold under additional constraints or for specific cases. For instance, while Euler's conjecture failed broadly, subsequent work established that four fifth powers suffice in general, refining the understanding of Diophantine equations. Philosophically, disproofs provide absolute certainty within the given axiomatic framework, contrasting with proofs that may depend on unproven assumptions or incomplete verifications, thus emphasizing the asymmetry in mathematical validation where falsification is more straightforward than confirmation.^[41]

Conditional Results

Conditional results represent partial resolutions to conjectures through proofs that rely on the assumption of other unresolved conjectures or additional axioms, typically expressed in the form "If Conjecture C holds, then property P is true." These results bridge the gap between fully open problems and complete resolutions by establishing logical dependencies and consequences under hypothetical conditions. In mathematics, such conditional theorems are particularly prevalent in fields like analytic number theory, where assumptions about the distribution of zeros of L-functions enable sharper estimates and deeper insights into arithmetic phenomena.^[43] Among the types of conditional results, implications between conjectures stand out, where the truth of one open problem entails the truth of another, thereby creating chains of dependency that highlight interconnections across mathematical domains. Another common type involves conditional theorems in analytic number theory, such as those derived under the Generalized Riemann Hypothesis (GRH), which posits that all non-trivial zeros of Dirichlet L-functions lie on the critical line with real part 1/2. For instance, GRH implies effective versions of classical results like the prime number theorem in arithmetic progressions with improved error terms, advancing understanding of prime distributions without requiring the full resolution of the hypothesis itself. These implications not only test the plausibility of the assumed conjecture but also reveal structural relationships, such as how zero-free regions in L-functions influence growth rates of arithmetic functions.^[43] The value of conditional results lies in their ability to incrementally expand mathematical knowledge, providing verifiable consequences that motivate further research into the underlying assumptions and often serve as stepping stones toward unconditional proofs. By demonstrating what would follow from a conjecture's truth, they offer evidence of consistency and inspire targeted efforts to verify the hypothesis, as seen in the numerous number-theoretic advances predicated on GRH. Historically, in the 1940s, André Weil established proofs for the analogue of the Riemann hypothesis in the case of algebraic curves over finite fields, which formed a foundational special case for his broader conjectures on zeta functions of varieties, illustrating how conditional or partial approaches can illuminate general frameworks.^[44] Despite their contributions, conditional results have inherent limitations: they do not affirm the original conjecture and remain contingent on the unproven assumption, potentially leaving the core problem unresolved while only partially testing its internal logic and compatibility with established theory. This provisional nature underscores their role as tools for exploration rather than final settlements, emphasizing the need for eventual unconditional verification to fully integrate the derived insights into mathematics.^[43]

Independence from Axioms

In mathematical logic, a conjecture is said to be independent of a given axiomatic system if it can neither be proved nor disproved using the axioms and rules of inference of that system, assuming the system is consistent. This phenomenon arises when the conjecture is true in some models of the axioms but false in others, highlighting the limitations of formal systems in capturing all mathematical truths. Independence results often rely on advanced techniques such as forcing in set theory, where new models are constructed to satisfy or violate the conjecture, demonstrating that no derivation within the original axioms can resolve it. A foundational result establishing the possibility of independence is Kurt Gödel's first incompleteness theorem from 1931, which proves that in any consistent formal system capable of expressing basic arithmetic, there exist statements that are true but unprovable within the system. This theorem implies that certain conjectures may inherently escape proof or disproof, as they transcend the expressive power of the axioms. Building on this, Paul Cohen's 1963 work using forcing methods showed the independence of the Continuum Hypothesis (CH) from Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), the standard axiomatic foundation for most mathematics. CH, which posits that there is no set whose cardinality is strictly between that of the integers and the real numbers, was the first major conjecture proven independent of its axiomatic framework, resolved as such in the 1960s through Cohen's innovative technique of generic extensions. Model theory and forcing are key methods for establishing independence. In model theory, different models of the same axioms can interpret the conjecture differently, revealing its undecidability; for instance, forcing constructs a model of ZFC where CH fails by adding generic subsets to the universe of sets. These approaches underscore the incompleteness of axiomatic systems, prompting mathematicians to explore alternative axioms, such as those involving large cardinals (e.g., measurable cardinals), which can imply the negation of certain independent conjectures like CH in extended frameworks. The implications extend to the philosophy of mathematics, questioning whether all truths are axiomatizable and influencing the development of set-theoretic multiverse views, where multiple consistent universes coexist without a universal resolution.

Notable Mathematical Examples

Resolved Conjectures

Resolved conjectures represent pivotal achievements in mathematics, where long-standing hypotheses have been affirmatively proven or refuted, often through innovative techniques that advance broader fields. From the 17th century onward, resolutions have grown in complexity, incorporating advanced tools like elliptic curves, Ricci flow, and computational verification, reflecting the evolving sophistication of mathematical proof methods. These successes not only confirm or deny specific statements but also catalyze developments in related areas, such as the theory of modular forms and geometric analysis. In number theory, Fermat's Last Theorem, proposed by Pierre de Fermat in 1637, asserts that no positive integers a, b, and c satisfy a^n + b^n = c^n for any integer n > 2. Andrew Wiles proved the theorem in 1994 by establishing the modularity of semistable elliptic curves over the rationals, linking elliptic curves to modular forms via the Taniyama-Shimura conjecture (now theorem in this case). This resolution, detailed in his seminal paper, not only settled a 350-year-old problem but also propelled the Langlands program forward, deepening connections between number theory and representation theory. The proof's impact extended to modular forms, inspiring generalizations like the full modularity theorem proved in 2001, which resolved broader questions about elliptic curves.^[45] Geometry and topology feature prominent resolved conjectures, including the Poincaré Conjecture, stated by Henri Poincaré in 1904, which posits that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. Grigori Perelman provided a proof in a series of preprints from 2002 to 2003, employing Ricci flow with surgery to deform manifolds and demonstrate topological equivalence. His work built on Richard Hamilton's program, overcoming singularities through novel entropy functionals, and was verified by the mathematical community by 2006. Perelman was awarded the Fields Medal in 2006 for this achievement, though he declined it, and the Millennium Prize in 2010, which he also refused.^[46] The proof's influence spurred advances in geometric analysis and low-dimensional topology, enabling classifications of 3-manifolds via the geometrization conjecture, which Perelman proved simultaneously. Another geometric resolution is the Kepler Conjecture, formulated by Johannes Kepler in 1611, claiming that the face-centered cubic lattice achieves the maximum density for equal sphere packings in three-dimensional Euclidean space, approximately \pi / (3\sqrt{2}) \approx 0.7405. Thomas Hales proved this in 1998 through an exhaustive computational enumeration of possible configurations, reducing the problem to checking finitely many cases via linear programming and interval arithmetic. Although initial verification faced challenges due to the proof's computational scale, a formalization in 2014 using HOL Light confirmed its correctness. This resolution validated Kepler's intuition after nearly four centuries and influenced packing problems in higher dimensions and materials science.^[47] In graph theory, the Four Color Theorem, conjectured by Francis Guthrie in 1852, states that any planar map can be colored with at most four colors such that no adjacent regions share the same color. Kenneth Appel and Wolfgang Haken established this in 1976 via a computer-assisted proof, discharging over 1,200 reducible configurations to show no counterexample exists for minimal planar graphs. This was the first major theorem relying heavily on computation, sparking debates on proof verification but ultimately accepted after rigorous checking. The theorem's proof advanced algorithmic graph theory and inspired formal verification methods in mathematics.

Unresolved Conjectures

The unresolved conjectures in mathematics represent some of the most profound open questions, many of which have resisted proof or disproof for over a century despite extensive efforts. Among the most prominent are the six unsolved Millennium Prize Problems designated by the Clay Mathematics Institute in 2000, each carrying a $1 million prize for resolution; these include the Riemann Hypothesis, the P versus NP problem, and the Birch and Swinnerton-Dyer Conjecture, which remain unsolved as of 2025.^[8] Other notable examples, such as the Collatz Conjecture, continue to attract computational and theoretical scrutiny, highlighting the persistence of these challenges in number theory, complexity theory, and algebraic geometry. Ongoing collaborative initiatives, including Polymath projects, have advanced partial understanding but yielded no full resolutions.^[8] The Riemann Hypothesis, proposed by Bernhard Riemann in 1859, asserts that all non-trivial zeros of the Riemann zeta function \zeta(s) lie on the critical line where the real part of s is $1/2.^[23] This conjecture has far-reaching implications for the distribution of prime numbers, as it would refine the error term in the Prime Number Theorem. Partial progress includes the verification that the first $10^{36} non-trivial zeros lie on this line through extensive computations.^[48] Additionally, equivalent formulations exist, and special cases have been proven, such as the hypothesis holding for zeta functions over finite fields, as established by André Weil in the 1940s.^[49] Its status as a Millennium Problem underscores its centrality, with no resolution despite computational evidence supporting it up to extraordinarily high heights.^[8] The P versus NP problem, formulated by Stephen Cook in 1971, asks whether every problem whose solution can be verified quickly by a deterministic Turing machine (in polynomial time, class P) can also be solved quickly (class NP). Resolving this would revolutionize algorithms, cryptography, and optimization, potentially collapsing complexity classes and enabling efficient solutions to NP-complete problems like the traveling salesman problem. Partial results include barriers to proof techniques, such as relativization (Baker, Gill, Solovay, 1975) and natural proofs (Razborov, Rudich, 1997), which show that common methods cannot separate P from NP. No full resolution exists as of 2025, though the consensus among experts favors P ≠ NP, with implications for secure computing if proven otherwise.^[8] The Collatz Conjecture, introduced by Lothar Collatz in 1937, posits that for any positive integer n, iteratively applying the rule—if n is even, divide by 2; if odd, replace with $3n+1—will eventually reach 1. This simple iterative process has been computationally verified for all starting values up to $2^{71} (approximately $2.36 \times 10^{21}), yet no general proof exists.^[50] Significant partial progress came from Terence Tao in 2019, who showed that almost all positive integers satisfy a weakened version leading to small values, using ergodic theory and probabilistic methods.^[51] Its unresolved nature persists, with ongoing efforts focusing on cycle detection and density arguments, though it lacks a Millennium prize. The Birch and Swinnerton-Dyer Conjecture, developed in the 1960s by Bryan Birch and Peter Swinnerton-Dyer through computational experiments on elliptic curves, states that for an elliptic curve E over the rationals, the rank of the group of rational points E(\mathbb{Q}) equals the order of the zero of the L-function L(E,s) at s=1. This links analytic properties of L-functions to the arithmetic of elliptic curves, with applications to Diophantine equations and the rank conjecture. Partial results include proofs for ranks 0 and 1 (Kolyvagin, 1989; Gross-Zagier, 1986), and the full conjecture holds weakly (the leading term matches) under the Generalized Riemann Hypothesis. As a Millennium Problem, it remains open in 2025, with recent advances in modular forms providing equivalent formulations but no complete proof.^[8]

Applications Beyond Mathematics

In Physical Sciences

In the physical sciences, a conjecture refers to an unproven theoretical proposition that proposes a framework for understanding natural phenomena, often serving as a hypothesis awaiting empirical validation or theoretical refinement. Unlike mathematical conjectures, which rely on logical deduction, physical conjectures are shaped by both theoretical models and observational data, frequently addressing gaps in established theories like general relativity or quantum mechanics. For instance, in string theory, the landscape conjecture posits that the theory admits an enormous number of possible vacuum states, estimated at approximately $10^{500}, each corresponding to different physical laws and constants, as proposed by Leonard Susskind to explain the fine-tuning of our universe through anthropic selection.^[52] Prominent examples illustrate the role of conjectures in tackling longstanding puzzles. The black hole information paradox, conjectured by Stephen Hawking in the mid-1970s, suggested that quantum information is lost during black hole evaporation via Hawking radiation, challenging unitarity in quantum mechanics; this has been partially addressed through the holographic principle, which encodes information on the black hole's event horizon.^[53] Another example is Modified Newtonian Dynamics (MOND), introduced by Mordehai Milgrom in 1983 as an alternative to dark matter, modifying Newton's laws at low accelerations to explain galactic rotation curves without invoking unseen mass. These conjectures drive theoretical progress by offering testable predictions, such as MOND's implications for wide binary stars or cluster dynamics.^[54] Conjectures in physics often guide experimental design and evolve with accumulating data. In particle physics, supersymmetry conjectures, which posit partner particles for each Standard Model particle to stabilize the Higgs mass, have motivated searches at the Large Hadron Collider (LHC), where experiments like those from the ATLAS collaboration have set stringent limits on supersymmetric particle masses without detecting them, prompting refinements to the theory.^[55] A foundational conjecture bridging quantum field theory and gravity is the gauge/gravity duality, proposed by Juan Maldacena in 1997, which equates a gravitational theory in anti-de Sitter space to a conformal field theory on its boundary, enabling non-perturbative calculations in strongly coupled systems. However, physical conjectures face challenges related to falsifiability, particularly in regimes beyond current experimental reach, such as the Planck scale or cosmic inflation epochs, where predictions may be untestable with foreseeable technology, raising debates on their scientific status.^[56] This issue is acute in multiverse scenarios implied by string theory landscapes, where diverse vacua are observationally indistinguishable, yet such conjectures remain valuable for conceptual unification and inspiring indirect tests through cosmology or particle accelerators.^[57]

In Computer Science and Logic

In computer science and logic, conjectures serve as foundational assumptions that guide the development of algorithms, complexity classifications, and theoretical frameworks, often assuming barriers to efficient computation to motivate practical designs. For instance, the Exponential Time Hypothesis (ETH), introduced in the early 2000s, posits that 3-SAT, an NP-complete problem, cannot be solved in subexponential time, implying that many NP-hard problems resist significantly faster than brute-force exponential algorithms. This conjecture, building on ideas from the 1990s, underpins numerous lower bound proofs in parameterized complexity and exact algorithms, preventing overly optimistic runtime improvements.^[58] In logic and computability theory, conjectures highlight undecidability and growth rates beyond recursive functions. The independence of the Axiom of Choice from ZF set theory extends to computability contexts, where its adoption affects the definability of choice functions in recursive mathematics; for example, in reverse mathematics, AC is equivalent to certain comprehension axioms over recursive sets, but its full strength remains independent of basic computable principles.^[59] Similarly, the Busy Beaver function, which measures the maximum steps a halting n-state Turing machine can take, grows faster than any computable function, leading to conjectures about its values, such as the determination in 2024 that BB(5) = 47,176,870 using the Coq proof assistant, fueling debates on halting problem variants and non-computable bounds.^[60] These examples underscore how conjectures delineate the limits of mechanical reasoning in logic. Conjectures also drive algorithm design, particularly in approximation and space-bounded computation. In approximation algorithms, hardness conjectures, like those assuming NP-hardness of achieving better than a constant factor for problems such as the traveling salesman, justify the development of polynomial-time heuristics with guaranteed ratios, as seen in seminal works on combinatorial optimization.^[61] Variants of hierarchy separations, such as the conjecture that P ≠ PSPACE—believed due to the exponential space needs for problems like quantified Boolean formulas—inform space-efficient algorithm paradigms and interactive proof systems.^[62] The Collatz conjecture, in a computational context, motivates termination analysis techniques, such as modeling it as string rewriting systems where automated provers seek invariants to confirm convergence for all starting integers, though full proof remains elusive.^[63] Quantum computing introduces conjectures separating BQP from P, positing that quantum polynomial-time machines solve problems like factoring efficiently while classical ones do not, relative to oracles that preserve this gap.^[64] Post-2000 developments have expanded this landscape, with quantum conjectures exploring BQP's position in the polynomial hierarchy and machine learning learnability hypotheses, such as those on the hardness of agnostic learning parities under cryptographic assumptions, shaping robust AI models by assuming limits on generalization from data.^[65] These advancements, intertwined with logic's foundational role in verifying computational claims, continue to propel interdisciplinary progress.^[66]

References

[1]
[PDF] Lecture 16 : Definitions, theorems, proofs Meanings Examples
Conjecture: A statement believed to be true, but for which we have no proof. (a statement that is being proposed to be a true statement). • Axiom: A basic ...
[2]
Definitions, Theorems, and Conjectures
Conjectures: Sometimes, there are statements in math that people believe to be true, but no one has been able to prove them yet. These are called conjectures. 🔗.
[3]
[PDF] On Mathematical Conjectures and Counterexamples
Jan 1, 2019 · These include the twin prime conjecture (the con- jecture that there are an infinite number of twin primes) and the Goldbach conjecture the ...
[4]
Topics: Mathematical Conjectures
* History: Proposed in 1742 by Prussian mathematician Christian Goldbach; 1996, Checked up to 2 × 106; The U.S. publisher of Uncle Petros and Goldbach's ...
[5]
Numb3rs 106: Prime Suspect - Cornell Mathematics
As was made obvious in the episode, the Riemann Hypothesis is one of the most famous conjectures in mathematics. It was originally stated in an 1859 paper ...<|control11|><|separator|>
[6]
Poincaré Conjecture - Clay Mathematics Institute
The Poincaré Conjecture asked if the 3D sphere is the unique simply connected 3-manifold, similar to a rubber band on an apple, not a doughnut.
[7]
[PDF] Million Buck Problems - Department of Mathematics
Poincaré conjecture, and the Goldbach Conjecture were all famous open problems. Ten years later, the Four Color Problem and the Alexandrov Conjecture were ...
[8]
The Millennium Prize Problems - Clay Mathematics Institute
The Prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium.Navier-Stokes Equation · Poincaré Conjecture · P vs NP · Rules
[9]
Birch and Swinnerton-Dyer Conjecture - Clay Mathematics Institute
The conjecture relates points on an elliptic curve to the rank of rational points. If ζ(1) = 0, there are infinite points; if not, finite.
[10]
Hodge Conjecture - Clay Mathematics Institute
Hodge Conjecture. The answer to this conjecture determines how much of the topology of the solution set of a system of algebraic equations can be defined in ...
[11]
[PDF] The Millennium Prize Problems - Clay Mathematics Institute
MILLENNIUM PRIZE PROBLEMS. 3. The Conjecture and Attempts to Prove It. Most complexity theorists believe that P 6= NP. Perhaps this can be partly explained by ...
[12]
Conjecture -- from Wolfram MathWorld
A proposition which is consistent with known data, but has neither been verified nor shown to be false. It is synonymous with hypothesis.
[13]
Conjectures | Brilliant Math & Science Wiki
A conjecture is a mathematical statement that has not yet been rigorously proved. Conjectures arise when one notices a pattern that holds true for many cases.
[14]
Axiom -- from Wolfram MathWorld
An axiom is a proposition regarded as self-evidently true without proof. The word "axiom" is a slightly archaic synonym for postulate.
[15]
Theorem -- from Wolfram MathWorld
A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments. In general, a theorem is an embodiment of some ...
[16]
The Subtle Art of the Mathematical Conjecture - Quanta Magazine
May 7, 2019 · It's an educated guess, not a proof. But a good conjecture will guide math forward, pointing the way into the mathematical unknown.
[17]
Goldbach Conjecture -- from Wolfram MathWorld
The conjecture that all odd numbers >=9 are the sum of three odd primes is called the "weak" Goldbach conjecture.
[18]
Conjecture - Etymology, Origin & Meaning
From late 14c. Old French and Latin origin, conjecture means interpretation or guess based on signs or evidence, and also to infer or predict by forming ...
[19]
Ben's observation - Math Central
... Pythagoras (500 BC). Their observation would have been based on the fact that the sum of consecutive odd numbers is a perfect square: 1, 1+3=4, 1+3+5 = 9, 1 ...
[20]
Euclid's Elements, Book IX, Proposition 20 - Clark University
This proposition states that there are more than any finite number of prime numbers, that is to say, there are infinitely many primes. Outline of the proof.Missing: unproven statements
[21]
Tracing the early history of algebra: Testimonies on Diophantus in ...
The aim of this paper is to investigate testimonies and other piece of evidence suggesting that Diophantus' heritage was present in intellectual milieus of the ...
[22]
Fermat's Last Theorem proof secures mathematics' top prize for Sir ...
Mar 15, 2016 · First formulated by the French mathematician Pierre de Fermat in 1637, it states: There are no whole number solutions to the equation xn + y ...
[23]
Riemann Hypothesis - Clay Mathematics Institute
Formulated in Riemann's 1859 paper, it asserts that all the 'non-obvious' zeros of the zeta function are complex numbers with real part 1/2.Riemann's 1859 Manuscript · The millennium prize problems I
[24]
[PDF] Mathematical Problems
A reprint of appears in Mathematical Developments Arising from Hilbert Problems edited by Felix. Brouder, American Mathematical Society, 1976. The original ...<|separator|>
[25]
[PDF] Mathematical Induction and Induction in Mathematics
Mathematical induction is a deductive technique, unlike empirical induction, which leads to probable, not necessarily true, conclusions.
[26]
Twin Primes -- from Wolfram MathWorld
It is conjectured that every even number is a sum of a pair of twin primes except a finite number of exceptions whose first few terms are 2, 4, 94, 96, 98, 400, ...
[27]
AMS :: Feature Column :: Why Do We Expect Lots of Twin Primes?
Heuristic reasoning about twin primes. Twin primes are a pair p p , p+2 p + 2 which are both primes. For example, from one of the plots above we see the twin ...<|separator|>
[28]
Conjecture and hypothesis: The importance of reality checks - PMC
Mar 28, 2017 · Conjecture is an idea, hypothesis is a conjecture that can be tested by experiment or observation, and consensus emerges when other interested colleagues agree.
[29]
https://www.ams.org/publicoutreach/feature-column/fc-2014-11
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[PDF] The Terminology of a Math Paper
It is often used like an informal lemma. • Conjecture. A statement that is unproved, but is believed to be true (Collatz conjecture, Goldbach conjecture,.
[31]
Set Theory - Stanford Encyclopedia of Philosophy
Oct 8, 2014 · Set theory is the mathematical theory of well-determined collections, called sets, of objects that are called members, or elements, of the set.
[32]
Gödel's Incompleteness Theorems
Nov 11, 2013 · The article was published in January 1931 (Gödel 1931; helpful introductions to Gödel's original paper are Kleene 1986 and Zach 2005). The ...
[33]
[PDF] 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove ...
Direct Proof: Assume p, and then use the rules of inference, axioms, defi- nitions, and logical equivalences to prove q. • Indirect Proof or Proof by ...Missing: conjectures | Show results with:conjectures
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[PDF] Proofs and Mathematical Reasoning - University of Birmingham
Direct proof assumes a given hypothesis, or any other known statement, and then logically deduces a conclusion. Indirect proof, also called proof by ...
[35]
Advanced Higher Maths Methods of Proof
Direct proof: proving A ⟹ B by assuming ... Contradiction: proving a conjecture by assuming its negation and showing that it leads to an absurdity ...<|control11|><|separator|>
[36]
Advancing mathematics by guiding human intuition with AI - Nature
Dec 1, 2021 · The practice of mathematics involves discovering patterns and using these to formulate and prove conjectures, resulting in theorems.
[37]
Strange new universes: Proof assistants and synthetic foundations
Feb 15, 2024 · Existing computer programs called proof assistants can verify the correctness of mathematical proofs but their specialized proof languages ...
[38]
[PDF] Reliably Reproducing Machine-Checked Proofs with the Coq Platform
The Coq proof assistant [10] provides a formal language to write datatypes, func- tions, and theorems, together with an environment for semi-interactive develop ...<|separator|>
[39]
Proof assistants | plus.maths.org
Aug 14, 2025 · This article explores how the simple notion of counting ends in a revolutionary new way of doing maths using proof assistants. This article is ...
[40]
https://plus.maths.org/content/sample-collection
[41]
[PDF] The Automation of Proof: A Historical and Sociological Exploration
This article reviews the history of the use of computers to auto- mate mathematical proofs. It identifies three broad strands of work: automatic theorem ...
[42]
[PDF] Problems of the Millennium: the Riemann Hypothesis
On the other hand, many deep results in number theory which are consequences of a general Riemann hypothesis can be shown to hold independently of it, thus ...
[43]
[PDF] The Riemann Hypothesis over Finite Fields - James Milne
Weil's work on the Riemann hypothesis for curves over finite fields led him to state his famous “Weil conjectures”, which drove much of the progress in.Missing: conditional | Show results with:conditional
[44]
The Core of Fermat's Last Theorem Just Got Superpowered
Jun 2, 2025 · This intermediate proof involved showing that an important kind of equation called an elliptic curve can always be tied to a completely ...
[45]
[PDF] Millennium Prize for the Poincaré Conjecture
Mar 18, 2010 · Perelman's breakthrough proof of the Poincaré conjecture was made possible by a number of new elements. He achieved a complete understanding of ...
[46]
[PDF] A proof of the Kepler conjecture - Annals of Mathematics
By a packing, we mean an arrangement of congruent balls that are nonover- lapping in the sense that the interiors of the balls are pairwise disjoint. Con-. Page ...
[47]
Riemann Zeta Function Zeros -- from Wolfram MathWorld
Gourdon (2004) used an algorithm of Odlyzko and Schönhage to calculate the first 10×10^(12) zeros (Pegg 2004, Pegg and Weisstein 2004). The following table ...
[48]
Andrew Odlyzko: Papers on Zeros of the Riemann Zeta Function ...
Andrew Odlyzko: Papers on Zeros of the Riemann Zeta Function and Related Topics (see also "Tables of zeros of the zeta function" and "Some unpublished materials ...
[49]
Mathematician Proves Huge Result on 'Dangerous' Problem
Dec 11, 2019 · Mathematicians regard the Collatz conjecture as a quagmire and warn each other to stay away. But now Terence Tao has made more progress than ...
[50]
[hep-th/0302219] The Anthropic Landscape of String Theory - arXiv
Feb 27, 2003 · I discuss the theoretical and conceptual issues that arise in developing a cosmology based on the diversity of environments implicit in string theory.
[51]
The Black Hole Paradox Nears Its End - UC Berkeley Physics
Mar 23, 2021 · In a landmark series of calculations, physicists have proved that black holes can shed information, which seems impossible by definition.
[52]
https://arxiv.org/abs/hep-th/0302219
[53]
Broken symmetry: searches for supersymmetry at the LHC
Dec 8, 2017 · A commentary by ATLAS physicists Paul de Jong and George Redlinger on the history, progress and future of the search for supersymmetry.
[54]
https://ui.adsabs.harvard.edu/abs/2002ARA&A..40..263S/abstract
[55]
https://atlas.cern/updates/feature/supersymmetry
[56]
[PDF] Lower bounds based on the Exponential Time Hypothesis
In this article we survey algorithmic lower bound results that have been obtained in the field of exact exponential time algorithms and pa- rameterized ...
[57]
The Axiom of Choice in Computability Theory and Reverse ... - arXiv
May 30, 2020 · Abstract:The Axiom of Choice (AC for short) is the most (in)famous axiom of the usual foundations of mathematics, ZFC set theory.Missing: independence | Show results with:independence
[58]
[PDF] The Busy Beaver Frontier - Scott Aaronson
It's natural to ask if there are functions that grow even faster than Busy Beaver—let's say “much” faster, in the sense that they still couldn't be computably ...
[59]
[PDF] Approximation Algorithms
In Part I we cover a combinato- rial algorithms for a number of important problems, using a wide variety of algorithm design techniques. ... conjectures ...
[60]
[PDF] Some Estimated Likelihoods For Computational Complexity
Nevertheless, P = PSPACE does look extremely unlikely: the idea that PSPACE corresponds to computing winning strategies in two-player games makes it clear.
[61]
[2105.14697] An Automated Approach to the Collatz Conjecture - arXiv
May 31, 2021 · We explore the Collatz conjecture and its variants through the lens of termination of string rewriting. We construct a rewriting system that ...<|separator|>
[62]
[PDF] BQP and the Polynomial Hierarchy - Scott Aaronson
A central task of quantum computing theory is to understand how BQP—meaning Bounded-Error Quantum Polynomial-Time, the class of all problems feasible for a ...
[63]
[PDF] arXiv:2201.07372v2 [cs.LG] 13 Jul 2023
Jul 13, 2023 · However, learning paradigms besides PAC learning and online learning lack formal and universally agreed-upon definitions of learnability.
[64]
Symbioses between mathematical logic and computer science
Mathematical logic provides tools for understanding and unifying topics in computer science, while computer science provides new ways of looking at logical ...