Problems

63 of 63 problems

• AL002The Zariski cancellation problemIf A[x] is a polynomial ring, must A be one?open
• AN002The Riemann hypothesisEvery non-trivial zero of the zeta function lies on the line Re(s) = 1/2.open
• AN003The Littlewood conjectureFor any two reals, n·‖nα‖·‖nβ‖ gets arbitrarily close to zero.open
• AN009The invariant subspace problemDoes every bounded operator on a Hilbert space have a non-trivial closed invariant subspace?open
• AN010Schanuel's conjectureA transcendence statement that implies almost everything else in the subject.open
• AN011Sendov's conjectureIf a polynomial's roots lie in the unit disc, is every root near a critical point?open
• AX0002Ringel's Conjecture and Kotzig's Conjecture for large $n$For all sufficiently large $n$, the complete graph $K_{2n+1}$ decomposes into $2n+1$ edge-disjoint copies of any tree $T$ with $n$ edges. A "copy" of $T$ is the image $T.\text{map}(f_i)$ of $T$ under a vertex embedding $f_i : V \hookrightarrow \text{Fin}(2n+1)$; the copies are pairwise edge-disjoint and together cover every edge of $K_{2n+1}$. This follows from `kotzig_conjecture_large`; see `Papeopen
• CB002The sunflower conjectureDoes a bounded number of k-element sets force three of them to overlap in exactly the same core?open
• CB004The union-closed sets conjectureIn a family of sets closed under unions, is some element in at least half the sets?open
• CB005The lonely runner conjectureRunners at distinct speeds on a circular track: is each of them, at some moment, far from all the others?open
• CB006Singmaster's conjectureIs there a bound on how often a number appears in Pascal's triangle?open
• CB007The Erdős–Turán conjecture on additive basesIf every number is a sum of two members of a set, must some number have many such representations?open
• CB008The Erdős conjecture on arithmetic progressionsIf the reciprocals of a set diverge, does it contain arbitrarily long progressions?open
• CX001The fundamental theorem of statistical learningA class of binary classifiers can be learned from finitely many samples exactly when its VC dimension is finite.◆open
• CX003Do the two definitions of finite VC dimension agree?A bridge between the corpus's two ways of saying a concept class has bounded shattering.open
• EP0003Erdős Problem 1041**Erdős–Herzog–Piranian Component Lemma** (Metric Properties of Polynomials, 1958): If $f$ is a monic degree $n$ polynomial with all roots in the unit disk, then some connected component of $\{z \mid |f(z)| < 1\}$ contains at least two roots with multiplicity. See p. 139, above Problem 5: [EHP58] Erdős, P. and Herzog, F. and Piranian, G., _Metric properties of polynomials_. J. Analyse Math. (1958)open
• EP0005Erdős Problem 1094For all $n\ge 2k$ the least prime factor of $\binom{n}{k}$ is $\le\max(n/k,k)$, with only finitely many exceptions.open
• EP0010Erdős Problem 248Are there infinitely many $n$ such that $\omega(n + k) \ll k$ for all $k \geq 1$? Here $\omega(n)$ is the number of distinct prime divisors of $n$.open
• EP0014Erdős Problem 266Let $a_n$ be an infinite sequence of positive integers such that $\sum \frac{1}{a_n}$ converges. There exists some integer $t \ge 1$ such that $\sum \frac{1}{a_n + t}$ is irrational. This was disproven by Kovač and Tao in [KoTa24]. [KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series. [arXiv:2406.17593](https://arxiv.org/abs/2406.17593) (2024).open
• EP0032Erdős Problem 865There exists a constant $C>0$ such that, for all large $N$, if $A\subseteq \{1,\ldots,N\}$ has size at least $\frac{5}{8}N+C$ then there are distinct $a,b,c\in A$ such that $a+b,a+c,b+c\in A$. A problem of Erdős and Sós (also earlier considered by Choi, Erdős, and Szemerédi [CES75], but Erdős had forgotten this).open
• EP0037Erdős Problem 946There are infinitely many $n$ such that $τ(n) = τ(n+1)$. Proved in [He84]. Here τ is the divisor counting function, which is `σ 0` in mathlib.open
• EP0038Erdős Problem 952Is there an infinite sequence of distinct Gaussian primes $x_1,x_2,\ldots$ such that $\lvert x_{n+1}-x_n\rvert \ll 1$?open
• GT001How many colours does the plane need?Colour every point of the plane so no two points exactly one unit apart share a colour. Five is not enough; seven suffice. Is six?open
• GT002Is the Ramsey number R(5,5) equal to 43?How many people must be at a party before five of them are all mutual friends or all mutual strangers? Known to be between 43 and 46.open
• GT003The Erdős–Gyárfás conjectureDoes every graph with minimum degree three contain a cycle of length a power of two?open
• GT004Hadwiger's conjectureIf a graph has no K_t minor, is it (t−1)-colourable?open
• GT005The reconstruction conjectureIs a graph determined by its vertex-deleted subgraphs?open
• GT006The cycle double cover conjectureDoes every bridgeless graph have a family of cycles covering each edge exactly twice?open
• GT007The graceful tree conjectureCan every tree be labelled so its edge differences are all distinct?open
• JC002The plane Jacobian conjectureDoes every polynomial map ℂ² → ℂ² with constant non-zero Jacobian determinant have a polynomial inverse? Open since 1939, and still open after the dimension-3 case was refuted in July 2026.◆open
• JC003The Jacobian conjecture fails in dimension 3A polynomial map ℂ³ → ℂ³ whose Jacobian determinant is a non-zero constant need not be invertible. Refute the 1939 Jacobian conjecture in dimension three by exhibiting such a map that is not injective.◆solved
• MO0001Mathoverflow 507128There exists a proper ideal `I` in a (commutative) total ring `R` of fractions that is an invertible module. If `I ⊊ R` is such an example, `I` must have infinite order in the Picard group, and `R` must not be Noetherian (otherwise it must be semi-local and therefore have trivial Picard group).open
• NT001There are infinitely many primesFor every natural number n there exists a prime p with p ≥ n. A warm-up problem: short to state, short to prove, and a good way to check your setup works.◆solved
• NT003Are there infinitely many twin primes?Pairs of primes two apart — 11 and 13, 29 and 31 — appear to keep occurring forever, but nobody can prove they do.open
• NT004Does every number reach one under the Collatz map?Halve if even, triple and add one if odd. Every number tried so far eventually reaches 1. Nobody can prove they all do.open
• NT005Is every even number the sum of two primes?4 = 2+2, 6 = 3+3, 8 = 3+5. Checked past 4·10^18, unproved since 1742.open
• NT006The abc conjectureIf a + b = c with a, b coprime, then c cannot be much larger than the product of the distinct primes dividing abc.open
• NT007Is the totient series irrational?Sum φ(n)/2ⁿ over all n, where φ counts integers coprime to n. Is the result irrational?open
• NT012Are there odd perfect numbers?A perfect number equals the sum of its proper divisors. Every one known is even. Nobody can rule out an odd one.open
• NT013Legendre's conjectureIs there always a prime between consecutive squares?open
• NT014The Erdős–Straus conjectureCan 4/n always be written as a sum of three unit fractions?open
• NT015Are there infinitely many Mersenne primes?Primes of the form 2^p − 1. Fifty-two are known. Whether the list ends is unknown.open
• NT016Brocard's problemIs n! + 1 a perfect square only for n = 4, 5 and 7?open
• NT020Are there infinitely many primes of the form n²+1?The fourth of Landau's problems, and like the other three, untouched.open
• NT021Lehmer's totient problemDoes φ(n) divide n − 1 only when n is prime?open
• NT022Carmichael's totient conjectureIs every value of Euler's totient attained at least twice?open
• NT023Andrica's conjectureIs the difference of square roots of consecutive primes always less than one?open
• NT024The Agoh–Giuga conjectureIs n prime exactly when the (n−1)-st powers of 1…n−1 sum to −1 mod n?open
• NT025Firoozbakht's conjectureIs the n-th root of the n-th prime strictly decreasing?open
• NT026Waring's problem for fourth powersIs every sufficiently large number a sum of sixteen fourth powers?open
• PA0004Gourevitch's series identityThe Gourevitch series identity: The following idenitity holds: $\sum_{n=0}^{\infty} \frac{1 + 14 n + 76 n^2 + 168 n^3}{2^{20 n}} \binom{2n}{n}^7 = \frac{32}{\pi^3}.$ This was originally conjectured in [G2003] by Guillera and proven in [A2025] by Au.open
• WP0002Beal conjectureThe **Beal Conjecture**: if we are given positive integers $A, B, C, x, y, z$ such that $x, y, z > 2$ and $A^x + B^y = C^z$ then $A, B, C$ have a common divisor.open
• WP0003Beck–Fiala theorem and conjecture**The Beck–Fiala theorem** If $S_1, \dots, S_m \subseteq [n]$ is a set system of degree at most $t$, i.e. every $j \in [n]$ lies in at most $t$ of the sets, and $t \ge 1$, then there is a colouring $\chi \colon [n] \to \{-1, +1\}$ with $\left|\sum_{j \in S_i} \chi(j)\right| \le 2t - 1$ for every $i$. The hypothesis $t \ge 1$ is necessary: a system of degree $0$ consists of empty sets only, whose dopen
• WP0008Class number problem for real quadratic fieldsThere are infinitely many real quadratic fields `ℚ(√d)` with class number one, where `d > 1` is a squarefree integer.open
• WP0013Fibonacci PrimesThere are infinitely many Fibonacci primes, i.e., Fibonacci numbers that are prime It is also a barrier to defining a benchmark from this paper: https://arxiv.org/html/2505.13938v1 (see Figure 8).open
• WP0016Hall's conjectureHow close can a perfect square get to a perfect cube? Hall conjectured that $|y^2 - x^3| > C\,|x|^{1/2}$ for some constant $C>0$ and all integers with $y^2 \ne x^3$ — so the gap cannot be much smaller than the square root of $x$.open
• WP0020Kaplansky's Conjectures**The zero-divisor conjecture** If `G` is torsion-free, then the group algebra `K[G]` has no non-trivial zero divisors.open
• WP0021Komlós conjecture**The Komlós conjecture** There exists a universal constant $K > 0$ such that for all $n, m \in \mathbb{N}$ and all vectors $v_1, \dots, v_n \in \mathbb{R}^m$ with $\|v_i\|_2 \le 1$ (encoded here as $\sum_j v_{ij}^2 \le 1$), there exist signs $\varepsilon_i \in \{-1, +1\}$ such that $\left\|\sum_i \varepsilon_i v_i\right\|_\infty \le K$, i.e. $\left|\sum_i \varepsilon_i v_{ij}\right| \le K$ for evopen
• WP0022Lander, Parkin, and Selfridge ConjectureThe Lander–Parkin–Selfridge conjecture: if the sum of $n$ positive integer $k$-th powers equals the sum of $m$ positive integer $k$-th powers, with all values on the left distinct from all values on the right, then $n + m \geq k$. Formally, for positive integers $k, n, m \in \mathbb{N}$ and sequences $x : \{0, \ldots, n-1\} \to \mathbb{N}$ and $y : \{0, \ldots, m-1\} \to \mathbb{N}$ with $x_i > 0$open
• WP0028Ramanujan τ-functionThe Ramanujan-Petersson conjecture: $|\tau(p)| \le 2 p^{11/2}$ for primes $p$.open
• WP0030Particular values of the Riemann zeta function$\zeta(5)$ is irrational.open
• WP0034Six standard deviations suffice (Spencer's theorem)**Six standard deviations suffice** (Spencer, 1985) For every $n$ and every family of $n$ subsets $S_1, \dots, S_n$ of $\{1, \dots, n\}$, there is a colouring $\chi : \{1, \dots, n\} \to \{-1, +1\}$ such that $\left|\sum_{j \in S_i} \chi(j)\right| \le 6\sqrt{n}$ for every $i$.open
• WP0040Are there infinitely many Woodall primes?A Woodall number has the form $k \cdot 2^k - 1$. Are infinitely many of them prime? Only a few dozen are known, the largest with millions of digits.open