This is a list of notable mathematical conjectures .
The following conjectures remain open. The (incomplete) column "cites" lists the number of results for a Google Scholar search for the term, in double quotes as of September 2022 [update] .
The conjecture terminology may persist: theorems often enough may still be referred to as conjectures, using the anachronistic names.
Priority date [13] | Proved by | Former name | Field | Comments |
---|---|---|---|---|
1962 | Walter Feit and John G. Thompson | Burnside conjecture that, apart from cyclic groups, finite simple groups have even order | finite simple groups | Feit–Thompson theorem⇔trivially the "odd order theorem" that finite groups of odd order are solvable groups |
1968 | Gerhard Ringel and John William Theodore Youngs | Heawood conjecture | graph theory | Ringel-Youngs theorem |
1971 | Daniel Quillen | Adams conjecture | algebraic topology | On the J-homomorphism, proposed 1963 by Frank Adams |
1973 | Pierre Deligne | Weil conjectures | algebraic geometry | ⇒Ramanujan–Petersson conjecture Proposed by André Weil. Deligne's theorems completed around 15 years of work on the general case. |
1975 | Henryk Hecht and Wilfried Schmid | Blattner's conjecture | representation theory for semisimple groups | |
1975 | William Haboush | Mumford conjecture | geometric invariant theory | Haboush's theorem |
1976 | Kenneth Appel and Wolfgang Haken | Four color theorem | graph colouring | Traditionally called a "theorem", long before the proof. |
1976 | Daniel Quillen; and independently by Andrei Suslin | Serre's conjecture on projective modules | polynomial rings | Quillen–Suslin theorem |
1977 | Alberto Calderón | Denjoy's conjecture | rectifiable curves | A result claimed in 1909 by Arnaud Denjoy, proved by Calderón as a by-product of work on Cauchy singular operators [14] |
1978 | Roger Heath-Brown and Samuel James Patterson | Kummer's conjecture on cubic Gauss sums | equidistribution | |
1983 | Gerd Faltings | Mordell conjecture | number theory | ⇐Faltings's theorem, the Shafarevich conjecture on finiteness of isomorphism classes of abelian varieties. The reduction step was by Alexey Parshin. |
1983 onwards | Neil Robertson and Paul D. Seymour | Wagner's conjecture | graph theory | Now generally known as the graph minor theorem. |
1983 | Michel Raynaud | Manin–Mumford conjecture | diophantine geometry | The Tate–Voloch conjecture is a quantitative (diophantine approximation) derived conjecture for p-adic varieties. |
c.1984 | Collective work | Smith conjecture | knot theory | Based on work of William Thurston on hyperbolic structures on 3-manifolds, with results by William Meeks and Shing-Tung Yau on minimal surfaces in 3-manifolds, also with Hyman Bass, Cameron Gordon, Peter Shalen, and Rick Litherland, written up by Bass and John Morgan. |
1984 | Louis de Branges de Bourcia | Bieberbach conjecture, 1916 | complex analysis | ⇐Robertson conjecture⇐Milin conjecture⇐de Branges's theorem [15] |
1984 | Gunnar Carlsson | Segal's conjecture | homotopy theory | |
1984 | Haynes Miller | Sullivan conjecture | classifying spaces | Miller proved the version on mapping BG to a finite complex. |
1987 | Grigory Margulis | Oppenheim conjecture | diophantine approximation | Margulis proved the conjecture with ergodic theory methods. |
1989 | Vladimir I. Chernousov | Weil's conjecture on Tamagawa numbers | algebraic groups | The problem, based on Siegel's theory for quadratic forms, submitted to a long series of case analysis steps. |
1990 | Ken Ribet | epsilon conjecture | modular forms | |
1992 | Richard Borcherds | Conway–Norton conjecture | sporadic groups | Usually called monstrous moonshine |
1994 | David Harbater and Michel Raynaud | Abhyankar's conjecture | algebraic geometry | |
1994 | Andrew Wiles | Fermat's Last Theorem | number theory | ⇔The modularity theorem for semistable elliptic curves. Proof completed with Richard Taylor. |
1994 | Fred Galvin | Dinitz conjecture | combinatorics | |
1995 | Doron Zeilberger [16] | Alternating sign matrix conjecture, | enumerative combinatorics | |
1996 | Vladimir Voevodsky | Milnor conjecture | algebraic K-theory | Voevodsky's theorem, ⇐norm residue isomorphism theorem⇔Beilinson–Lichtenbaum conjecture, Quillen–Lichtenbaum conjecture. The ambiguous term "Bloch-Kato conjecture" may refer to what is now the norm residue isomorphism theorem. |
1998 | Thomas Callister Hales | Kepler conjecture | sphere packing | |
1998 | Thomas Callister Hales and Sean McLaughlin | dodecahedral conjecture | Voronoi decompositions | |
2000 | Krzysztof Kurdyka, Tadeusz Mostowski, and Adam Parusiński | Gradient conjecture | gradient vector fields | Attributed to René Thom, c.1970. |
2001 | Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor | Taniyama–Shimura conjecture | elliptic curves | Now the modularity theorem for elliptic curves. Once known as the "Weil conjecture". |
2001 | Mark Haiman | n! conjecture | representation theory | |
2001 | Daniel Frohardt and Kay Magaard [17] | Guralnick–Thompson conjecture | monodromy groups | |
2002 | Preda Mihăilescu | Catalan's conjecture, 1844 | exponential diophantine equations | ⇐Pillai's conjecture⇐abc conjecture Mihăilescu's theorem |
2002 | Maria Chudnovsky, Neil Robertson, Paul D. Seymour, and Robin Thomas | strong perfect graph conjecture | perfect graphs | Chudnovsky–Robertson–Seymour–Thomas theorem |
2002 | Grigori Perelman | Poincaré conjecture, 1904 | 3-manifolds | |
2003 | Grigori Perelman | geometrization conjecture of Thurston | 3-manifolds | ⇒spherical space form conjecture |
2003 | Ben Green; and independently by Alexander Sapozhenko | Cameron–Erdős conjecture | sum-free sets | |
2003 | Nils Dencker | Nirenberg–Treves conjecture | pseudo-differential operators | |
2004 (see comment) | Nobuo Iiyori and Hiroshi Yamaki | Frobenius conjecture | group theory | A consequence of the classification of finite simple groups, completed in 2004 by the usual standards of pure mathematics. |
2004 | Adam Marcus and Gábor Tardos | Stanley–Wilf conjecture | permutation classes | Marcus–Tardos theorem |
2004 | Ualbai U. Umirbaev and Ivan P. Shestakov | Nagata's conjecture on automorphisms | polynomial rings | |
2004 | Ian Agol; and independently by Danny Calegari–David Gabai | tameness conjecture | geometric topology | ⇒Ahlfors measure conjecture |
2008 | Avraham Trahtman | Road coloring conjecture | graph theory | |
2008 | Chandrashekhar Khare and Jean-Pierre Wintenberger | Serre's modularity conjecture | modular forms | |
2009 | Jeremy Kahn and Vladimir Markovic | surface subgroup conjecture | 3-manifolds | ⇒Ehrenpreis conjecture on quasiconformality |
2009 | Jeremie Chalopin and Daniel Gonçalves | Scheinerman's conjecture | intersection graphs | |
2010 | Terence Tao and Van H. Vu | circular law | random matrix theory | |
2011 | Joel Friedman; and independently by Igor Mineyev | Hanna Neumann conjecture | group theory | |
2012 | Simon Brendle | Hsiang–Lawson's conjecture | differential geometry | |
2012 | Fernando Codá Marques and André Neves | Willmore conjecture | differential geometry | |
2013 | Yitang Zhang | bounded gap conjecture | number theory | The sequence of gaps between consecutive prime numbers has a finite lim inf. See Polymath Project#Polymath8 for quantitative results. |
2013 | Adam Marcus, Daniel Spielman and Nikhil Srivastava | Kadison–Singer problem | functional analysis | The original problem posed by Kadison and Singer was not a conjecture: its authors believed it false. As reformulated, it became the "paving conjecture" for Euclidean spaces, and then a question on random polynomials, in which latter form it was solved affirmatively. |
2015 | Jean Bourgain, Ciprian Demeter, and Larry Guth | Main conjecture in Vinogradov's mean-value theorem | analytic number theory | Bourgain–Demeter–Guth theorem, ⇐ decoupling theorem [18] |
2018 | Karim Adiprasito | g-conjecture | combinatorics | |
2019 | Dimitris Koukoulopoulos and James Maynard | Duffin–Schaeffer conjecture | number theory | Rational approximation of irrational numbers |
The conjectures in following list were not necessarily generally accepted as true before being disproved.
In mathematics, ideas are supposedly not accepted as fact until they have been rigorously proved. However, there have been some ideas that were fairly accepted in the past but which were subsequently shown to be false. The following list is meant to serve as a repository for compiling a list of such ideas.
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's Last Theorem, have shaped much of mathematical history as new areas of mathematics are developed in order to prove them.
A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.
In mathematics, a transcendental number is a real or complex number that is not algebraic – that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best-known transcendental numbers are π and e.
In mathematics, a theorem is a statement that has been proved, or can be proved. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems.
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "student John Smith is not lazy" is a counterexample to the generalization "students are lazy", and both a counterexample to, and disproof of, the universal quantification "all students are lazy."
The modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms. Andrew Wiles and Richard Taylor proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's Last Theorem. Later, a series of papers by Wiles's former students Brian Conrad, Fred Diamond and Richard Taylor, culminating in a joint paper with Christophe Breuil, extended Wiles's techniques to prove the full modularity theorem in 2001.
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof, which proves the existence of a particular kind of object without providing an example. For avoiding confusion with the stronger concept that follows, such a constructive proof is sometimes called an effective proof.
In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers.
The Gelfond–Schneider constant or Hilbert number is two to the power of the square root of two:
Transcendental number theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.
In mathematics, specifically transcendental number theory, Schanuel's conjecture is a conjecture made by Stephen Schanuel in the 1960s concerning the transcendence degree of certain field extensions of the rational numbers.
In mathematics, a proof of impossibility is a proof that demonstrates that a particular problem cannot be solved as described in the claim, or that a particular set of problems cannot be solved in general. Such a case is also known as a negative proof, proof of an impossibility theorem, or negative result. Proofs of impossibility often are the resolutions to decades or centuries of work attempting to find a solution, eventually proving that there is no solution. Proving that something is impossible is usually much harder than the opposite task, as it is often necessary to develop a proof that works in general, rather than to just show a particular example. Impossibility theorems are usually expressible as negative existential propositions or universal propositions in logic.
In number theory, the Pólya conjecture stated that "most" of the natural numbers less than any given number have an odd number of prime factors. The conjecture was set forth by the Hungarian mathematician George Pólya in 1919, and proved false in 1958 by C. Brian Haselgrove. Though mathematicians typically refer to this statement as the Pólya conjecture, Pólya never actually conjectured that the statement was true; rather, he showed that the truth of the statement would imply the Riemann hypothesis. For this reason, it is more accurately called "Pólya's problem".
In number theory, the Chinese hypothesis is a disproven conjecture stating that an integer n is prime if and only if it satisfies the condition that is divisible by n—in other words, that an integer n is prime if and only if . It is true that if n is prime, then , however the converse is false, and therefore the hypothesis as a whole is false. The smallest counterexample is n = 341 = 11×31. Composite numbers n for which is divisible by n are called Poulet numbers. They are a special class of Fermat pseudoprimes.
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions.
In mathematics, Carmichael's totient function conjecture concerns the multiplicity of values of Euler's totient function φ(n), which counts the number of integers less than and coprime to n. It states that, for every n there is at least one other integer m ≠ n such that φ(m) = φ(n). Robert Carmichael first stated this conjecture in 1907, but as a theorem rather than as a conjecture. However, his proof was faulty, and in 1922, he retracted his claim and stated the conjecture as an open problem.
Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were believed to be impossible to prove using current knowledge by almost all contemporary mathematicians.
Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is his "revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite. This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument. The title of the article, "On a Property of the Collection of All Real Algebraic Numbers", refers to its first theorem: the set of real algebraic numbers is countable. Cantor's article was published in 1874. In 1879, he modified his uncountability proof by using the topological notion of a set being dense in an interval.
The Lander, Parkin, and Selfridge conjecture concerns the integer solutions of equations which contain sums of like powers. The equations are generalisations of those considered in Fermat's Last Theorem. The conjecture is that if the sum of some k-th powers equals the sum of some other k-th powers, then the total number of terms in both sums combined must be at least k.
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