Unsolved Mathematical Conjectures Rankings
A ranking of the 50 most prestigious open problems in mathematics, from the Millennium Prize enigmas to notorious number theory traps.
Science & TechnologyTop 50
No rankings yet. Be the first!
Similar Lists
All Items (50)
Riemann Hypothesis β All non-trivial zeros of the zeta function have a real part of exactly one-half (1859)β
P versus NP Problem β Determining whether every problem whose solution can be quickly verified can also be quickly solved (1971)β
Navier-Stokes Existence and Smoothness β Proving that solutions to the fluid dynamics equations always exist and do not blow up (2000)β
Birch and Swinnerton-Dyer Conjecture β Relating the number of points on an elliptic curve to the behavior of its L-function (1965)β
Hodge Conjecture β Asserting that certain de Rham cohomology classes are algebraic in complex projective varieties (1950)β
Yang-Mills Existence and Mass Gap β Proving the rigorous mathematical foundation of the strong nuclear force and its mass gap (1954)β
Goldbach's Conjecture β Every even integer greater than 2 can be expressed as the sum of two primes (1742)β
Collatz Conjecture (3n + 1) β Proving that iterating n/2 for evens and 3n+1 for odds always eventually reaches 1 (1937)β
Twin Prime Conjecture β There are infinitely many prime numbers that differ by exactly 2 (1846)β
abc Conjecture β Bounding the prime factors of c relative to a and b where a + b = c (1985)β
Beal Conjecture β A generalization of Fermat's Last Theorem for equations A^x + B^y = C^z with a common prime factor (1993)β
ErdΕs-Straus Conjecture β The fraction 4/n can always be expressed as the sum of three distinct unit fractions (1948)β
Littlewood Conjecture β Bounding the simultaneous Diophantine approximation of two real numbers (1930)β
Lonely Runner Conjecture β Out of k runners on a circular track, every runner will eventually be distance 1/k from all others (1967)β
Jacobian Conjecture β A polynomial mapping with a non-zero constant Jacobian determinant is invertible (1939)β
Inverse Galois Problem β Determining whether every finite group appears as the Galois group of some extension of the rationals (19th Century)β
Novikov Conjecture β The topological invariance of higher signatures in algebraic topology (1970)β
Whitehead Conjecture β Every subcomplex of an aspherical two-dimensional complex is itself aspherical (1941)β
Standard Conjectures on Algebraic Cycles β Foundational assumptions needed to complete Grothendieck's theory of motives (1968)β
Tate Conjecture β Characterizing algebraic cycles on a variety in terms of Galois representations (1965)β
Lehmer's Totient Problem β Asking if there is any composite number n such that Euler's totient function Ο(n) divides n - 1 (1932)β
Carmichael's Totient Conjecture β If Ο(x) = m has one solution, it must have at least one other solution (1922)β
Brocard's Problem β Finding integer values where n! + 1 = m^2, currently only known for n = 4, 5, and 7 (1876)β
Gilbreath's Conjecture β The forward difference sequence of primes always begins with 1 (1958)β
Singmaster's Conjecture β Bounding the number of times a number other than 1 can appear in Pascal's Triangle (1971)β
Elliott-Halberstam Conjecture β Bounding the error term in the distribution of primes in arithmetic progressions (1968)β
Hardy-Littlewood Prime Tuple Conjecture β Predicting the asymptotic density of specific constellations of prime numbers (1923)β
Bunyakovsky Conjecture β A polynomial of degree greater than 1 yields infinitely many primes under specific conditions (1857)β
Bateman-Horn Conjecture β A broad generalization of prime frequency distributions encompassing Twin Prime and Bunyakovsky (1962)β
Schinzel's Hypothesis H β Predicting when a finite set of polynomials will simultaneously generate prime values (1958)β
Artin's Conjecture on Primitive Roots β Predicting the frequency with which a given integer is a primitive root modulo primes (1927)β
Catalan-Dickson Conjecture on Aliquot Sequences β Every aliquot sequence either terminates at zero or settles into a repeating cycle (1843)β
Perfect Cuboid Existence β Finding a rectangular cuboid where all edges, face diagonals, and the space diagonal are integers (1719)β
Hadwiger's Conjecture β Any graph requiring k colors must contain a complete graph of order k as a minor (1943)β
Kissing Number Problem (High Dimensions) β Determining the exact kissing numbers for dimension 5, 6, 7, and beyond (1638)β
Chromatic Number of the Plane β Determining the exact number of colors needed to color the plane without connecting identical colors at distance 1 (1950)β
ErdΕs-GyΓ‘rfΓ‘s Conjecture β Every graph with minimum degree 3 contains a cycle whose length is a power of 2 (1995)β
Vizing's Planar Graph Conjecture β The edge chromatic number of a planar graph with maximum degree 6 is exactly 6 (1965)β
Graceful Tree Conjecture β Every tree graph can be labeled 'gracefully' with sequential vertex and edge differences (1967)β
Reconstruction Conjecture β Any graph with at least 3 vertices is uniquely determined by its vertex-deleted subgraphs (1941)β
List Coloring Conjecture β The list chromatic index of any multigraph is equal to its edge chromatic index (1985)β
Union-Closed Sets Conjecture β Any family of sets closed under union has an element contained in at least half the sets (1979)β
Schanuel's Conjecture β Generalizing algebraic independence for exponential functions over complex numbers (1960)β
Leopoldt's Conjecture β Bounding the p-adic rank of the unit group in algebraic number fields (1962)β
Farrell-Jones Conjecture β Formulating the algebraic K-theory of group rings in geometric topology (1993)β
Baum-Connes Conjecture β Relating the K-theory of C*-algebras to the K-homology of classifying spaces (1982)β
Zariski Conjecture β Relating the singularities of plane algebraic curves to their topological embedding (1929)β
Montgomery's Pair Correlation Conjecture β The normalized spacing of zeta zeros matches the eigenvalues of random Hermitian matrices (1973)β
Iwaniec's Conjecture β Bounding the spectral gaps of the Laplacian on modular surfaces (1982)β
Generalized Riemann Hypothesis β Extending the Riemann Hypothesis to all Dirichlet L-functions (1884)β