Great problems

Ancient

1. 100
   Odd perfect numbersArithmetic and number theory. The Euclid–Euler theorem completely characterizes even perfect numbers as $2^{p-1}(2^p - 1)$ where $2^p - 1$ is a Mersenne prime, but whether an odd perfect number can exist is one of the oldest open problems in mathematics. Any hypothetical example must take Euler's form $N = q^k m^2$ with $q \equiv k \equiv 1 \pmod{4}$, exceed $10^{1500}$, and have at least 10 distinct prime factors.Open

Early modern

1. 1640
   Finiteness of Fermat primesArithmetic and number theory. Fermat conjectured in 1640 that every number $2^{2^n}+1$ is prime; Euler refuted this in 1732 by factoring $F_5$. Since then every tested Fermat number from $F_5$ through $F_{32}$ has turned out composite, and no sixth Fermat prime has ever been found, leading to the modern (opposite) conjecture that only finitely many exist.Open
2. 1644
   Infinitude of Mersenne primesArithmetic and number theory. Marin Mersenne listed exponents he believed gave primes $2^p-1$ in 1644. Only 52 Mersenne primes are known today, the largest, $2^{136279841}-1$, found in 2024 with over 41 million digits. Each new one is exponentially rarer to find, and although heuristic models predict infinitely many, no proof exists.Open
3. 1694
   Kissing number problemGeometry. How many equal non-overlapping spheres in $\mathbb{R}^n$ can simultaneously touch a central sphere of the same size? Originating in the 1694 dispute between Isaac Newton and David Gregory for $n=3$, the exact value $k(n)$ is known only in dimensions $n = 1, 2, 3, 4, 8, 24$.Partially solved

18th century

1. 1734
   Irrationality of the Euler–Mascheroni constantArithmetic and number theory, Analysis. Introduced by Leonhard Euler in 1734 as the limiting difference between the harmonic series $H_n = \sum_{k=1}^n 1/k$ and the natural logarithm $\ln n$, the constant $\gamma$ appears throughout analysis, number theory, and probability, notably as the constant term in the Laurent expansion $\zeta(s) = \frac{1}{s-1} + \gamma + O(s-1)$ at $s = 1$ and in the Weierstrass product for $\Gamma(z)$. Despite being one of the three classical constants of analysis alongside $\pi$ and $e$, it is still unknown whether $\gamma$ is rational, irrational, algebraic, or transcendental. Continued-fraction computations show that if $\gamma = p/q$ is rational, its denominator must satisfy $q > 10^{242080}$.Open
2. 1734
   Irrationality of ζ(5) and odd zeta valuesArithmetic and number theory. In 1734, Leonhard Euler evaluated $\zeta(2) = \pi^2/6$ and more generally showed that every even zeta value $\zeta(2k)$ is a rational multiple of $\pi^{2k}$, hence transcendental. By contrast, no closed formula in terms of $\pi$ is known for odd zeta values $\zeta(2k+1)$. Roger Apéry stunned the mathematical world in 1978 by proving that $\zeta(3)$ is irrational, and in 2000–2001 Tanguy Rivoal and Keith Ball proved that infinitely many odd zeta values are irrational, while Wadim Zudilin proved that at least one of $\zeta(5), \zeta(7), \zeta(9), \zeta(11)$ is irrational. Yet the irrationality of any specific value $\zeta(2k+1)$ with $k \ge 2$, starting with $\zeta(5)$, remains open.Open
3. 1742
   Goldbach conjectureArithmetic and number theory. In a 1742 letter to Euler, Goldbach proposed that every integer greater than 2 is a sum of three primes (counting 1 as prime, as was then customary); Euler reformulated the now-standard, stronger statement that every even integer greater than 2 is a sum of two primes. Despite being checked by computer for every even number up to $4\times10^{18}$, and despite strong partial results, the general statement remains unproved. A separate, weaker statement — the ternary (or weak) Goldbach conjecture, that every odd number greater than 5 is a sum of three primes — was fully proved by Helfgott in 2013.Landau #1Open

19th century

1. 1808
   Legendre's conjectureArithmetic and number theory. Asks whether there is always a prime between any two consecutive squares $n^2$ and $(n+1)^2$ — an interval of length $2n + 1 \approx 2x^{1/2}$ at $x = n^2$. Named by Edmund Landau in 1912 as one of four 'unattackable' prime problems, it remains open even assuming the Riemann hypothesis.Landau #3Open
2. 1834
   Gauss circle problemArithmetic and number theory, Analysis. Because each integer lattice point $(m,n) \in \mathbb{Z}^2$ is the center of a unit square of area $1$, the number of lattice points inside $x^2+y^2 \le r^2$ is approximated by the area $\pi r^2$. In 1834, Carl Friedrich Gauss proved the elementary geometric bound $|E(r)| \le 2\sqrt{2}\pi r$, showing $E(r) = O(r)$. Sierpiński improved this to $O(r^{2/3})$ in 1906, while G. H. Hardy and Edmund Landau independently proved in 1915 that $E(r)$ cannot be $O(r^{1/2})$, so $\theta \ge \tfrac{1}{2}$. Despite a century of exponential-sum refinements culminating in Martin Huxley's published bound $\theta \le \tfrac{131}{208} \approx 0.62981$, the conjectured value $\theta = \tfrac{1}{2}$ remains open.Open
3. 1849
   Dirichlet divisor problemArithmetic and number theory, Analysis. Counting the pairs of positive integers $(a,b)$ with $ab \le x$ is geometrically equivalent to counting integer lattice points under the rectangular hyperbola $ab = x$ in the first quadrant. By splitting the region at $\sqrt{x}$ (the Dirichlet hyperbola method), Peter Gustav Lejeune Dirichlet proved in 1849 that $D(x) = x\log x + (2\gamma - 1)x + O(x^{1/2})$. Voronoi lowered the error exponent to $\tfrac{1}{3}$ in 1904, and Hardy and Landau showed in 1915 that $\theta \ge \tfrac{1}{4}$. Because $\sum_{n=1}^{\infty} d(n)n^{-s} = \zeta(s)^2$, the problem is intimately tied to the growth of the Riemann zeta function on the critical line and shares its exponential-sum machinery with the Gauss circle problem, with Martin Huxley's 2003 bound $\theta \le \tfrac{131}{416} \approx 0.31490$ standing as the published record.Open
4. 1849
   Twin prime conjectureArithmetic and number theory. Pairs like $(3,5)$, $(11,13)$, $(17,19)$ are called twin primes. Alphonse de Polignac stated the conjecture in general form in 1849, as the case $n=2$ of the claim that every even $n$ is a gap between infinitely many consecutive primes. The problem was considered hopeless until Yitang Zhang proved in 2013 that some gap below a fixed bound recurs infinitely often — a first, stunning step toward the conjecture. Sieve-theoretic refinements have since brought the bound from 70{,}000{,}000 down to 246, but no method so far can force the gap all the way down to 2.Landau #2Open
5. 1859
   Riemann hypothesisArithmetic and number theory, Analysis. The zeta function encodes the distribution of prime numbers: its zeros control the error term in the prime counting function $\pi(x)$. Riemann observed in 1859 that all the nontrivial zeros he could compute lay on the critical line $\mathrm{Re}(s)=\tfrac12$ and conjectured this always holds. A proof would give the strongest possible form of the prime number theorem; billions of zeros have been checked numerically and all lie on the line, but no proof or disproof is known. It is one of the seven Clay Millennium Prize Problems.MillenniumHilbert #8Open
6. 1876
   Brocard's problemArithmetic and number theory. Posed by Henri Brocard in 1876 and again in 1885, and independently by Srinivasa Ramanujan in 1913, the problem asks when $n! + 1$ is a perfect square $m^2$. The pairs $(n, m)$ satisfying the equation are called Brown numbers; only three are known: $4! + 1 = 5^2$, $5! + 1 = 11^2$, and $7! + 1 = 71^2$. Paul Erdős conjectured that no further solutions exist. Assuming the $abc$ conjecture, Marius Overholt proved in 1993 that there are only finitely many solutions, and computer searches have checked up to $n = 10^{15}$ without finding a fourth solution.Open
7. 1878
   Continuum hypothesisFoundations of mathematics. Cantor showed that the reals are uncountable, so their cardinality $2^{\aleph_0}$ is strictly larger than that of the integers, $\aleph_0$. The continuum hypothesis (CH) asks whether $2^{\aleph_0}$ is the very next cardinal, $\aleph_1$. Gödel (1940) showed CH cannot be disproved from the standard ZFC axioms, and Cohen (1963) showed it cannot be proved either, using the new method of forcing. CH is therefore independent of ZFC: both CH and its negation are consistent with the usual axioms of set theory, so the question has no answer within ZFC alone.Hilbert #1KGKurt Gödel, 1963
8. 1882
   Irrationality of e + π and e·πArithmetic and number theory. Charles Hermite proved in 1873 that $e$ is transcendental, and Ferdinand von Lindemann proved in 1882 that $\pi$ is transcendental. Because $(x - e)(x - \pi) = x^2 - (e + \pi)x + e\pi$, the two numbers $e$ and $\pi$ are the roots of a quadratic polynomial whose coefficients are $-(e + \pi)$ and $e\pi$. Since the roots of a polynomial with algebraic coefficients are themselves algebraic, $e + \pi$ and $e\pi$ cannot both be algebraic: at least one of $e + \pi$ and $e\pi$ is transcendental (and hence irrational). Yet it remains completely unknown whether either individual number $e + \pi$ or $e\pi$ is even irrational.Open
9. 1892
   Inverse Galois problemAlgebra. Originating in 19th-century algebra and formalized by David Hilbert (1892) and Emmy Noether (1918), the Inverse Galois Problem asks whether every finite group is a Galois group over $\mathbb{Q}$. It is solved for all solvable groups (Shafarevich 1954) and all 26 sporadic simple groups, but remains open in general.Open
10. 1893
    Hadamard conjectureCombinatorics and discrete mathematics, Algebra. Does an $n \times n$ orthogonal matrix with $\pm 1$ entries exist for every multiple of $4$? Originating with Sylvester (1867), Hadamard's maximal determinant theorem (1893), and Paley's finite-field constructions (1933), the general conjecture remains open even as computer and AI-assisted searches have constructed Hadamard matrices for all multiples of $4$ up to $2000$.Open

20th century

1. 1900
   Hilbert's sixteenth problemDifferential equations and dynamical systems, Geometry. Asks for the possible arrangements of ovals of a degree-$n$ real algebraic curve and for a uniform bound $H(n)$ on the number of limit cycles of a degree-$n$ polynomial vector field in the plane. Every individual polynomial system has finitely many limit cycles (Ilyashenko 1991; Écalle 1992), but whether $H(n) < \infty$ remains open even for $n = 2$.Hilbert #16Smale #13Open
2. 1900
   Hilbert's sixth problemMathematical physics. Hilbert asked both for an axiomatic foundation of probability and mechanics and for a rigorous derivation of macroscopic fluid equations from microscopic particle collisions. Kolmogorov axiomatized probability in 1933, and in 2025 Deng, Hani, and Ma derived the long-time Boltzmann and fluid equations from hard-sphere dynamics, while axiomatizing all of modern physics remains open.Hilbert #6Partially solved
3. 1900
   Hilbert's twelfth problemArithmetic and number theory. Asks for explicit analytic generators of every abelian extension of a number field $K$, generalizing how roots of unity $\exp(2\pi i/n)$ generate $\mathbb{Q}^{\mathrm{ab}}$. Solved for imaginary quadratic fields by complex multiplication and for totally real fields in a $p$-adic sense by Dasgupta and Kakde (2021–2023), while the general complex-analytic problem remains open.Hilbert #12Open
4. 1902
   Burnside problemAlgebra. Posed by William Burnside in 1902, the problem split into three major branches: the General Burnside Problem is false (Golod–Shafarevich 1964), the Bounded Burnside Problem is true for $n \in \{1, 2, 3, 4, 6\}$ and false for large exponents (Novikov–Adian 1968; Ivanov 1994; Lysenok 1996) while remaining open for small exponents such as $n = 5$, and the Restricted Burnside Problem is true for all exponents (Zelmanov 1989–1990).Partially solved
5. 1904
   Infinitude of Sophie Germain primesArithmetic and number theory. A prime $p$ for which $2p+1$ is also prime is called a Sophie Germain prime, and the companion $2p+1$ is called a safe prime. Sophie Germain introduced these primes in the early nineteenth century to prove the first case of Fermat's Last Theorem for such exponents. While safe primes are widely used in public-key cryptography (such as Diffie–Hellman key exchange) and massive examples with hundreds of thousands of digits have been found computationally, the infinitude of Sophie Germain primes remains unproven. The conjecture is a special case of Dickson's conjecture and Schinzel's hypothesis H, and Hardy and Littlewood gave a precise heuristic asymptotic formula for their density.Open
6. 1908
   Lindelöf hypothesisArithmetic and number theory, Analysis. Formulated by the Finnish mathematician Ernst Leonard Lindelöf in 1908, the Lindelöf hypothesis asserts that on the critical line $\mathrm{Re}(s) = \tfrac{1}{2}$, the Riemann zeta function grows slower than any positive power of $t$. The Phragmén–Lindelöf convexity principle combined with the functional equation gives the baseline convexity bound $\mu(\tfrac{1}{2}) \le \tfrac{1}{4}$. While the Riemann hypothesis implies the Lindelöf hypothesis (and even the finer bound $|\zeta(\tfrac{1}{2}+it)| \ll \exp(C \log t / \log\log t)$), the Lindelöf hypothesis is already strong enough to imply the density hypothesis for zeros of $\zeta(s)$ and the asymptotic prime number theorem in short intervals $[x, x+x^{1/2+\varepsilon}]$.Open
7. 1909
   Normality of πArithmetic and number theory, Probability and statistics. Émile Borel introduced normal numbers in 1909 and proved using the Borel–Cantelli lemma that almost all real numbers (with respect to Lebesgue measure) are normal in every integer base $b \ge 2$. Yet proving normality for any naturally occurring constant not artificially constructed from digit concatenations remains extraordinarily difficult. For $\pi$, trillions of decimal and hexadecimal digits have been computed and pass statistical tests for uniform frequency $10^{-k}$, and the 1996 Bailey–Borwein–Plouffe (BBP) formula links base-$16$ normality of $\pi$ to equidistribution of a chaotic map, but it is still unknown even whether every digit $0, 1, \dots, 9$ appears infinitely often in the decimal expansion of $\pi$.Open
8. 1910
   Unknotting problem (complexity of recognising the unknot)Topology, Applied and computational mathematics. Proved decidable by Wolfgang Haken in 1961 and now known unconditionally to lie in $\mathsf{NP} \cap \mathsf{coNP}$ with a quasi-polynomial $n^{O(\log n)}$ algorithm announced by Marc Lackenby in 2021, while membership in $\mathsf{P}$ remains open.Partially solved
9. 1911
   Inscribed square problem (Toeplitz)Geometry, Topology. Does every continuous simple closed loop in the plane contain the four corners of a square? Proved for smooth, piecewise analytic, and locally monotone curves, and extended to all rectangles on smooth curves (Greene–Lobb, 2020), but still open for arbitrary continuous Jordan curves.Open
10. 1912
    Primes of the form n² + 1 (Landau's fourth problem)Arithmetic and number theory. Dirichlet's theorem guarantees infinitely many primes in any coprime linear progression $an + b$, yet no single-variable polynomial of degree two or higher has ever been proved to take infinitely many prime values. Posed by Edmund Landau in 1912 as the fourth of his problems on primes, the case $n^2 + 1$ remains open; the closest unconditional result is Iwaniec's 1978 theorem that $n^2 + 1$ is a product of at most two primes infinitely often.Landau #4Open
11. 1917
    Kakeya conjectureAnalysis, Geometry. A set in $\mathbb{R}^n$ containing a unit segment in every direction can have measure zero, but must have full Hausdorff dimension $n$: proved for $n=2$ (Davies, 1971) and in a landmark 2025 preprint for $n=3$ (Wang–Zahl), while $n \ge 4$ remains open.Partially solved
12. 1917
    No-three-in-line problemCombinatorics and discrete mathematics, Geometry. How many points can be placed on an $n \times n$ grid with no three on a line? By the pigeonhole principle at most $2$ points can lie in each row, so $f(n) \le 2n$; Henry Dudeney posed the $n = 8$ chessboard case in 1917, and while $f(n) = 2n$ holds for all tested $n \le 74$, the best general constructions for large $n$ achieve only $(3/2 - o(1))n$.Open
13. 1920
    Suslin's problemFoundations of mathematics. Published posthumously in 1920 by Mikhail Suslin, this question asks whether separability in Cantor's characterization of $(\mathbb{R}, <)$ can be weakened to the countable chain condition. Work by Jech, Tennenbaum, Jensen, and Solovay–Tennenbaum (1967–1971) established that Suslin's hypothesis is independent of ZFC.Independent of the axioms
14. 1927
    Artin's conjecture on primitive rootsArithmetic and number theory, Algebra. An integer $a$ is a primitive root modulo a prime $p$ if its powers generate the entire multiplicative group $(\mathbb{Z}/p\mathbb{Z})^{\times}$ of order $p-1$. In 1927, Emil Artin proposed to Helmut Hasse that any integer $a \ne -1$ that is not a square generates $(\mathbb{Z}/p\mathbb{Z})^{\times}$ for a positive proportion of all primes $p$, governed by an Euler product heuristic. Later computations by Lehmer revealed small entanglement corrections when the square-free part of $a$ is $1 \pmod{4}$, after which Heilbronn tightened the density formula. Christopher Hooley proved the corrected conjecture in 1967 assuming the Generalized Riemann Hypothesis, and D. R. Heath-Brown showed unconditionally in 1986 that at most two prime bases can fail — yet not a single specific base $a$ is unconditionally known to work.Open
15. 1930
    Littlewood conjectureArithmetic and number theory. In one dimension, badly approximable numbers $\alpha$ satisfy $\liminf_{n \to \infty} n \, \|n\alpha\| > 0$; quadratic irrationals such as $\sqrt{2}$ are classic examples. Around 1930, J. E. Littlewood conjectured that in two dimensions no such simultaneous barrier exists: multiplying the two approximation errors $\|n\alpha\|$ and $\|n\beta\|$ together always allows the product $n \, \|n\alpha\| \, \|n\beta\|$ to become arbitrarily small. In 2006, Manfred Einsiedler, Anatole Katok, and Elon Lindenstrauss proved using homogeneous dynamics that the set of possible counterexamples has Hausdorff dimension $0$, yet whether the exceptional set is completely empty remains open.Open
16. 1930
    Value of the Ramsey number R(5,5)Combinatorics and discrete mathematics. While $R(3, 3) = 6$ and $R(4, 4) = 18$ are known exactly, the fifth diagonal Ramsey number $R(5, 5)$ remains unknown after nearly a century of Ramsey theory, currently bracketed in the four-integer range $43 \le R(5, 5) \le 46$.Open
17. 1933
    Lehmer's Mahler measure problemArithmetic and number theory, Algebra. For a polynomial $P(x) = a_d \prod_{i=1}^d (x - \alpha_i)$ with integer coefficients, the Mahler measure is $M(P) = |a_d| \prod_{i=1}^d \max(1, |\alpha_i|)$. By Kronecker's theorem, $M(P) = 1$ precisely when $P(x)$ is a product of cyclotomic polynomials and powers of $x$. In 1933, D. H. Lehmer discovered a degree-$10$ polynomial with Mahler measure $\lambda \approx 1.1762808$ and asked whether any integer polynomial can have Mahler measure strictly between $1$ and $\lambda$. Despite strong lower bounds for special classes of polynomials and exhaustive computer searches up to degree $44$, the general question remains open.Open
18. 1935
    Happy ending problem (Erdős–Szekeres)Combinatorics and discrete mathematics, Geometry. How many points in general position in the plane are needed to guarantee a convex $n$-gon? Posed by Esther Klein for $n = 4$ in 1933—leading to her marriage to George Szekeres and Paul Erdős's nickname "the Happy Ending Problem"—the conjectured exact value $\mathrm{ES}(n) = 2^{n-2} + 1$ is proved only for $n \le 6$, though Andrew Suk established $\mathrm{ES}(n) = 2^{n + o(n)}$ in 2016.ErdősOpen
19. 1936
    Cramér's conjecture on prime gapsArithmetic and number theory, Probability and statistics. By the prime number theorem, the average gap between consecutive primes near $x$ is $\log x$. In 1936, Harald Cramér introduced a probabilistic model in which each integer $n \ge 3$ is declared 'prime' independently with probability $1/\log n$, and proved that with probability $1$ the maximal gaps in this random sequence grow like $(\log x)^2$. While Andrew Granville showed in 1995 that local divisibility by small primes suggests the limit superior should be at least $2e^{-\gamma} \approx 1.1229$ rather than $1$, the order-of-magnitude bound $g_n = O((\log p_n)^2)$ remains widely believed and completely beyond current methods — even the Riemann hypothesis only yields $g_n = O(\sqrt{p_n}\log p_n)$.Open
20. 1937
    Collatz conjectureArithmetic and number theory, Differential equations and dynamical systems. Repeatedly halve even numbers and triple-plus-one odd numbers: every positive integer is conjectured to eventually reach 1, though no proof exists despite decades of computer verification.Open
21. 1939
    Jacobian conjectureAlgebra, Analysis. Asks whether every polynomial map $k^n \to k^n$ in characteristic zero with constant nonzero Jacobian determinant has a polynomial inverse; disproved for $n \ge 3$ in 2026 while the planar case $n = 2$ remains open.Smale #16Partially solved
22. 1941
    Erdős–Turán conjecture on additive basesArithmetic and number theory, Combinatorics and discrete mathematics. If a set of natural numbers eventually covers all integers by pairwise sums (an asymptotic basis of order $2$), must some integers have arbitrarily many representations as a sum of two elements? Posed by Paul Erdős and Pál Turán in 1941, the conjecture $\limsup_{n \to \infty} r_A(n) = \infty$ remains open.ErdősOpen
23. 1941
    Graph reconstruction conjectureCombinatorics and discrete mathematics. Is every finite simple graph with at least $3$ vertices uniquely determined up to isomorphism by the multiset of its vertex-deleted subgraphs? Formulated by Paul J. Kelly and Stanisław Ulam in 1941, the conjecture is proved for trees, regular graphs, disconnected graphs, and almost all graphs, but remains open in general.Open
24. 1943
    Hadwiger's conjecture (graph minors)Combinatorics and discrete mathematics. Does $\chi(G) \ge t$ force $G$ to contain the complete graph $K_t$ as a minor? Posed by Hugo Hadwiger in 1943 as a sweeping generalization of the four-color theorem, it is proved for $t \le 6$ (Robertson, Seymour, and Thomas, 1993) and remains open for all $t \ge 7$.Open
25. 1946
    Erdős unit distance problemGeometry, Combinatorics and discrete mathematics. How many pairs among $n$ points in the plane can lie at distance $1$? Bounded above by $O(n^{4/3})$, while a 2026 number-theoretic construction disproved Erdős's $n^{1+o(1)}$ conjecture by proving $u(n) = \Omega(n^{1+c})$.ErdősOpen
26. 1948
    Erdős–Straus conjectureArithmetic and number theory. Formulated in 1948 by Paul Erdős and Ernst G. Straus, this Egyptian-fraction conjecture asks whether $4/n$ is always a sum of three reciprocals of positive integers. Because any decomposition for a divisor $d \mid n$ scales to a decomposition for $n$, it suffices to prove the statement when $n = p$ is prime. Simple algebraic identities immediately cover primes $p \not\equiv 1 \pmod{24}$, and larger modular covering systems eliminate almost all residue classes, enabling computer verification up to $n = 10^{17}$, yet a complete covering system of finitely many identities cannot exist.ErdősOpen
27. 1949
    Invariant subspace problemAnalysis. Asks whether every bounded linear operator on an infinite-dimensional separable complex Hilbert space has a nontrivial closed invariant subspace.Open
28. 1949
    Critical exponents of self-avoiding walksProbability and statistics. Introduced by Paul Flory in 1949 as a model of polymer chains, the self-avoiding walk is conjectured by physicists to have universal critical exponents ($\nu = 3/4$, $\gamma = 43/32$ in $d = 2$; $\nu \approx 0.5876$ in $d = 3$). Takashi Hara and Gordon Slade rigorously proved the mean-field values $\nu = 1/2$, $\gamma = 1$ for $d \ge 5$ in 1992 using the lace expansion, but the exponents remain unproven in the physically central dimensions $d = 2$ and $d = 3$.Open
29. 1950
    Hadwiger–Nelson problemCombinatorics and discrete mathematics, Geometry. How many colors are needed to color the plane $\mathbb{R}^2$ so that no two points at distance $1$ share a color? Posed by Edward Nelson in 1950 with classical bounds $4 \le \chi(\mathbb{R}^2) \le 7$, the lower bound was raised to $5$ by Aubrey de Grey in 2018, leaving $\chi(\mathbb{R}^2) \in \{5, 6, 7\}$ open.Open
30. 1950
    Heilbronn triangle problemGeometry. How large can the smallest triangle among $n$ points in the unit square be? Heilbronn's original $O(1/n^2)$ conjecture was disproved in 1981 ($\Delta(n) = \Omega(\log n / n^2)$), and in 2023–2024 Cohen, Pohoata, and Zakharov broke the 40-year-old upper bound record to prove $\Delta(n) \le n^{-7/6 + o(1)}$.Open
31. 1950
    Hodge conjectureGeometry, Algebra. Hodge classes are the cohomology classes that the complex structure of a variety picks out as 'shape-like'; the conjecture asks whether every one of them actually comes from a genuine algebraic subvariety. Known in codimension 1 since 1924 (Lefschetz) and for abelian varieties up to dimension 5, it is one of the seven Clay Millennium Prize Problems and remains open in general.MillenniumOpen
32. 1950
    Whitehead problemFoundations of mathematics, Algebra. Posed by J. H. C. Whitehead in 1950 in homological algebra, the question asks whether every abelian group $A$ with $\operatorname{Ext}^1_{\mathbb{Z}}(A, \mathbb{Z}) = 0$ is free. In 1974 Saharon Shelah proved that the answer is independent of ZFC: it is true in Gödel's constructible universe $V = L$ and false under Martin's Axiom plus $\neg\text{CH}$.Independent of the axioms
33. 1957
    Kaplansky's conjectures on group ringsAlgebra. Traced to Graham Higman's 1940 thesis and popularized by Irving Kaplansky at the 1956 Shelter Island conference, these conjectures propose that group rings of torsion-free groups are as structurally clean as integral domains. In February 2021, Giles Gardam disproved the unit conjecture using an explicit unit in $\mathbb{F}_2[P]$ for the Promislow group $P$, later extended to every characteristic; the zero-divisor and idempotent conjectures remain open in general but proven for many classes of groups.Partially solved
34. 1957
    Continuity of the phase transition in 3D percolationProbability and statistics. Broadbent and Hammersley introduced percolation in 1957 to model fluid flow through a random medium; whether the percolation probability $\theta(p)$ has no jump at the critical point $p_c$ (a continuous, 'second-order' phase transition) is proven in dimension $2$ and for $d \ge 11$ via the lace expansion, but remains a celebrated open problem for the physically relevant case $d = 3$.Open
35. 1958
    Schinzel's hypothesis H and Bunyakovsky's conjectureArithmetic and number theory. Viktor Bunyakovsky conjectured in 1857 that a single irreducible integer polynomial $f(x)$ with positive leading coefficient and no fixed prime divisor represents infinitely many primes; a century later, Andrzej Schinzel and Wacław Sierpiński extended this in 1958 to finite families of such polynomials (Hypothesis H). The only case completely proved is a single linear polynomial ($k=1, \deg f_1 = 1$), which is Dirichlet's theorem on arithmetic progressions. Hypothesis H unifies many of the deepest open problems in prime number theory: $f_1(x)=x, f_2(x)=x+2$ gives the twin prime conjecture, $f_1(x)=x, f_2(x)=2x+1$ gives the infinitude of Sophie Germain primes, and $f_1(x)=x^2+1$ gives Landau's fourth problem.Open
36. 1958
    Sendov's conjectureAnalysis. Asks whether every root of a complex polynomial in the closed unit disk lies within distance $1$ of a critical point; proved for degrees $n \le 8$ and for all sufficiently large degrees $n \ge n_0$.Partially solved
37. 1960
    Erdős–Rado sunflower conjectureCombinatorics and discrete mathematics. How large can a family of $w$-element sets be without containing $r$ sets whose pairwise intersections are all identical? Posed by Paul Erdős and Richard Rado in 1960 with the factorial bound $w! (r - 1)^w$, the problem saw a landmark breakthrough in 2019 by Alweiss, Lovett, Wu, and Zhang (refined to $(C r \log w)^w$), while the pure exponential bound $C_r^w$ remains open.ErdősOpen
38. 1961
    Smooth four-dimensional Poincaré conjectureTopology. The last remaining dimension of the generalized Poincaré conjecture across all categories: does the $4$-sphere $S^4$ admit any exotic smooth structures, or is every smooth homotopy $4$-sphere diffeomorphic to $S^4$?Open
39. 1963
    Tate conjectureArithmetic and number theory, Geometry. Proposed by John Tate at the 1963 Woods Hole conference, this arithmetic analogue of the Hodge conjecture asks whether every Galois-invariant $\ell$-adic cohomology class comes from an algebraic cycle. It is proven for divisors on abelian varieties over finite fields (Tate 1966) and number fields (Faltings 1983) and for K3 surfaces, but remains open for higher-codimension cycles on general varieties.Open
40. 1964
    Černý conjecture (synchronizing automata)Combinatorics and discrete mathematics, Applied and computational mathematics. If a finite automaton with $n$ states can be reset to a single state regardless of its starting state, how long must the shortest reset word be in the worst case? Ján Černý conjectured in 1964 that $(n-1)^2$ always suffices, yet the best known general upper bounds remain cubic in $n$.Open
41. 1965
    Birch and Swinnerton-Dyer conjectureArithmetic and number theory. Predicts that the rank of an elliptic curve's rational points equals the order of vanishing of its L-function at s=1 — proven only when that order is 0 or 1.MillenniumOpen
42. 1966
    Moser's worm problemGeometry. What is the smallest area of a planar shape that can cover every curve of length $1$ under rotation and translation? The minimum area remains open, bounded between $0.232239$ and $\pi/12 \approx 0.2618$ for convex covers (and $0.260437$ for non-convex covers).Open
43. 1966
    Moving sofa problemGeometry, Applied and computational mathematics. What is the largest two-dimensional shape that can maneuver around a right-angled corner in a hallway of width $1$? Gerver's 18-curve sofa achieves area $\approx 2.2195$, and a landmark November 2024 preprint by Jineon Baek claims a full proof of its optimality.Open
44. 1966
    Schanuel's conjectureArithmetic and number theory, Algebra. Formulated by Stephen Schanuel in a 1960s course at Yale and published in Serge Lang's Introduction to Transcendental Numbers (1966), Schanuel's conjecture is the central unifying conjecture of transcendental number theory for the exponential function. It subsumes the Lindemann–Weierstrass theorem (when $z_1, \dots, z_n$ are algebraic), Baker's theorem on linear independence of logarithms of algebraic numbers (when $e^{z_1}, \dots, e^{z_n}$ are algebraic), and the Gelfond–Schneider theorem, while also implying major open problems such as the algebraic independence of $e$ and $\pi$ and the transcendence of $e + \pi$, $e\pi$, $e^e$, and $\pi^e$. Only the case $n = 1$ (Hermite–Lindemann) and special cases where either all $z_i$ or all $e^{z_i}$ are algebraic are proved in full.Open
45. 1967
    Graceful tree conjectureCombinatorics and discrete mathematics. Can the vertices of every $m$-edge tree be labeled with distinct integers from $0$ to $m$ so that the absolute differences across its edges realize every integer from $1$ to $m$? Formulated by Ringel, Kotzig, and Rosa in the 1960s, the conjecture remains open despite verification up to $35$ vertices and the 2020 asymptotic solution of Ringel's decomposition conjecture.Open
46. 1967
    Langlands programArithmetic and number theory, Algebra. A grand unifying web of conjectures connecting number theory, representation theory, and algebraic geometry; proved for $\mathrm{GL}_n$ over function fields and in the unramified geometric setting (2024), while general reciprocity and functoriality over number fields remain open.Partially solved
47. 1967
    Lonely runner conjectureCombinatorics and discrete mathematics, Arithmetic and number theory. Posed independently by Jörg Wills (1967) and Thomas Cusick (1974), the lonely runner conjecture asks whether every runner on a circular track with distinct constant speeds is sometimes isolated from all others by a distance of at least $\frac{1}{n}$. The conjecture has been verified for $n \le 13$ runners through a series of breakthroughs from 2001 to 2026, but the general case remains open.Open
48. 1968
    Elliott–Halberstam conjectureArithmetic and number theory, Analysis. The Generalized Riemann Hypothesis predicts that primes up to $x$ are equidistributed among the $\varphi(q)$ coprime residue classes modulo $q$ with square-root error for every modulus $q \le x^{1/2-\varepsilon}$. In 1965, Enrico Bombieri and A. I. Vinogradov proved that this square-root level of distribution $\theta = 1/2$ holds unconditionally on average over $q$. In 1968, Peter Elliott and Heini Halberstam conjectured that on average over $q$, the level of distribution actually extends all the way to any $\theta < 1$ — beyond what even the Generalized Riemann Hypothesis gives for individual moduli. This conjecture is the key analytic input for modern sieve methods: Goldston, Pintz, and Yıldırım, and later Maynard and Polymath8, showed that it implies bounded gaps between primes down to $12$ (and $6$ under its generalized form).Open
49. 1969
    Grothendieck's standard conjectures on algebraic cyclesGeometry, Algebra. Formulated by Alexander Grothendieck in 1968 as a framework for a theory of motives, the standard conjectures propose deep structural properties linking algebraic cycles to cohomology. They are proven for curves, surfaces, abelian varieties, and certain hyperkähler manifolds, but remain open for general smooth projective varieties, even though Deligne's 1974 proof of the Weil conjectures bypassed them.Open
50. 1969
    Exponent of matrix multiplication $\omega$Applied and computational mathematics, Algebra. Since Strassen showed in 1969 that $\omega < \log_2 7 \approx 2.8074$, successive refinements of the laser method have pushed the upper bound down to $\omega < 2.371177$, while the trivial lower bound $\omega \ge 2$ remains unbroken; most researchers conjecture $\omega = 2$.Open
51. 1971
    Complexity of graph isomorphismApplied and computational mathematics, Combinatorics and discrete mathematics. Asks whether Graph Isomorphism can be solved in polynomial time ($\mathsf{P}$); László Babai proved in 2015–2017 that it is solvable in quasipolynomial time $\exp((\log n)^{O(1)})$.Open
52. 1971
    P versus NPApplied and computational mathematics, Foundations of mathematics. Asks whether every problem whose solution can be checked quickly can also be solved quickly — the central open question of computational complexity, carrying a Clay Millennium Prize.MillenniumOpen
53. 1973
    Erdős conjecture on arithmetic progressionsArithmetic and number theory, Combinatorics and discrete mathematics. If the sum of reciprocals $\sum_{n \in A} \frac{1}{n} = \infty$, must $A$ contain arbitrarily long arithmetic progressions? Proved for $k = 3$ by Thomas Bloom and Olof Sisask (2020), while all lengths $k \ge 4$ remain open.ErdősOpen
54. 1977
    Erdős–Hajnal conjectureCombinatorics and discrete mathematics. Whereas general $n$-vertex graphs need only contain a clique or independent set of logarithmic size $\Theta(\log n)$, Paul Erdős and András Hajnal conjectured in 1977 that forbidding any fixed induced subgraph $H$ forces a polynomial-size clique or independent set $n^{\delta_H}$.ErdősOpen
55. 1979
    Union-closed sets conjecture (Frankl)Combinatorics and discrete mathematics. In any finite union-closed family $\mathcal{F} \neq \{\emptyset\}$, must some element appear in at least $\frac{1}{2}$ of the sets? Posed by Péter Frankl in 1979, the conjecture saw a major breakthrough in 2022 when Justin Gilmer used Shannon entropy to prove the first constant lower bound ($0.01$, quickly sharpened to $\frac{3 - \sqrt{5}}{2} \approx 0.382$), while $\frac{1}{2}$ remains open.Open
56. 1982
    Local connectivity of the Mandelbrot set (MLC)Differential equations and dynamical systems, Analysis. Asks whether the Mandelbrot set $M \subset \mathbb{C}$ is locally connected—a conjecture that would give a complete combinatorial model of $M$ and imply the density of hyperbolic quadratic polynomials.Open
57. 1985
    abc conjectureArithmetic and number theory. Predicts that for coprime $a+b=c$, $c$ is rarely much larger than the product of the distinct primes dividing $abc$ — proposed in 1985; a disputed 2012 proof claim was published in 2021 but is not accepted by most number theorists.Open
58. 1993
    Beal conjectureArithmetic and number theory. Formulated in 1993 by Dallas banker and amateur mathematician Andrew Beal while investigating generalizations of Fermat's Last Theorem, the conjecture states that the generalized Fermat equation $A^x + B^y = C^z$ has no solutions in pairwise coprime positive integers $A, B, C$ when all three exponents satisfy $x, y, z \ge 3$. The condition $x, y, z > 2$ is sharp because $1^n + 2^3 = 3^2$ and identities such as $2^3 + 2^3 = 2^4$ (where the bases share the prime factor $2$) show that both the exponent threshold and the coprimality hypothesis are essential. The American Mathematical Society holds a USD $1{,}000{,}000$ prize funded by Beal for a proof or counterexample.Open
59. 1997
    Volume conjecture (knots)Topology. Proposed by Rinat Kashaev (1997) and Hitoshi and Jun Murakami (2001), the volume conjecture predicts that the asymptotic growth rate of the $N$-colored Jones polynomial at the root of unity $e^{2\pi i/N}$ recovers the hyperbolic volume of the knot complement.Open

Contemporary

1. 2000
   Navier–Stokes existence and smoothnessDifferential equations and dynamical systems. Asks whether the 3D incompressible Navier–Stokes equations always have smooth, globally defined solutions from smooth initial data, or whether they can blow up in finite time — one of the seven Clay Millennium Prize problems.MillenniumSmale #15Open
2. 2000
   Yang–Mills existence and mass gapMathematical physics, Analysis. Classical Yang–Mills fields are massless like the photon, yet in the quantum world the strong nuclear force they describe is short-range and binds gluons and quarks into massive particles. Giving a mathematically rigorous construction of four-dimensional quantum Yang–Mills theory and proving its mass gap $\Delta > 0$ is one of the seven Clay Millennium Prize Problems.MillenniumOpen
3. 2002
   Unique games conjectureApplied and computational mathematics. Proposed by Subhash Khot in 2002, the Unique Games Conjecture asserts that approximating a seemingly simple class of constraint satisfaction problems with bijective constraints is NP-hard; if true, it implies that standard semidefinite programming relaxations achieve the optimal approximation ratio for Max-Cut, Vertex Cover, and every constraint satisfaction problem.Open