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MOD-008 | O-Minimal Theory with Trans-Exponential Growth | Does there exist an o-minimal first-order theory with a trans-exponential (rapid growth) function? | O-minimal structures are ordered structures where definable sets have simple topology. Known o-minimal structures include real closed fields and structures with restricted analytic functions. The question asks whether o-minimality is compatible with very fast-growing functions, testing the limits of tame model theory.
... | open | L5: Millennium Prize | 5 | Algebra | null | null | null | 145 | 11 | train |
MOD-009 | Infinite Minimal Field Algebraic Closure | Is every infinite minimal field of characteristic zero algebraically closed? | A minimal structure is one where every definable subset is finite or cofinite. The question asks whether infinite fields with this property must be algebraically closed (when char=0). This would characterize the simplest infinite fields from a model-theoretic perspective, connecting field theory with minimality.
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MOD-010 | Keisler's Order | Determine the structure of Keisler's order on first-order theories. | Keisler's order compares first-order theories based on the complexity of their ultrapowers. Understanding this order would classify theories by their model-theoretic complexity. Recent breakthroughs have shed light on the order's structure, but a complete classification remains elusive. This connects with classificatio... | open | L5: Millennium Prize | 5 | Algebra | null | null | null | 156 | 12 | train |
ALG-013 | Serre's Conjecture II | For simply connected semisimple algebraic groups over fields of cohomological dimension ≤2, is $H^1(F,G) = 0$? | Serre's Conjecture II predicts that the first Galois cohomology of simply connected semisimple groups vanishes over fields of small cohomological dimension. This would have major implications for the classification of algebraic groups and forms. The conjecture is known for various classes of fields but remains open in ... | open | L5: Millennium Prize | 5 | Algebra | null | null | null | 178 | 14 | train |
ALG-015 | Uniform Boundedness Conjecture for Rational Points | Is there a bound N(g,d) such that all curves of genus g≥2 over degree d number fields have at most N(g,d) rational points? | The uniform boundedness conjecture asks whether the number of rational points on curves of genus ≥2 is uniformly bounded in terms of genus and field degree. This would be a remarkable strengthening of Faltings' theorem (finite number of points). The conjecture connects arithmetic geometry with Diophantine equations.
<... | open | L5: Millennium Prize | 5 | Algebra | null | null | null | 213 | 17 | train |
TOP-002 | Berge Conjecture | Are Berge knots the only knots in S³ admitting lens space surgeries? | The Berge conjecture states that Berge knots (constructed via a specific procedure) are the only knots in the 3-sphere that admit Dehn surgeries yielding lens spaces. This would classify all such knots, providing deep insight into the relationship between knot theory and 3-manifold topology.
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TOP-004 | Hilbert-Smith Conjecture | If a locally compact group acts faithfully and continuously on a manifold, must it be a Lie group? | The Hilbert-Smith conjecture asks whether every locally compact group with a continuous faithful action on a manifold is necessarily a Lie group. This would rule out p-adic groups acting on manifolds, resolving a fundamental question about the symmetries of topological spaces.
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## Liter... | open | L5: Millennium Prize | 5 | Topology | null | null | null | 212 | 17 | train |
TOP-008 | Whitehead Conjecture | Is every connected subcomplex of a 2-dimensional aspherical CW complex also aspherical? | The Whitehead conjecture asks whether asphericity (having contractible universal cover) is preserved under taking subcomplexes in dimension 2. This would clarify the local structure of aspherical spaces and has connections to group theory and low-dimensional topology.
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## Literature rev... | open | L4: Expert | 4 | Topology | null | null | null | 143 | 11 | train |
TOP-009 | Zeeman Conjecture | Is $K \times [0,1]$ collapsible for every finite contractible 2-dimensional CW complex K? | The Zeeman conjecture predicts that the product of any finite contractible 2-complex with an interval is collapsible (can be reduced to a point by elementary collapses). This relates to the Poincaré conjecture and questions about higher-dimensional manifolds. A counterexample would have major implications.
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COMB-001 | 1/3-2/3 Conjecture | Does every non-total finite poset have two elements x,y with P(x before y in random linear extension) ∈ [1/3, 2/3]? | The 1/3-2/3 conjecture asks whether finite partially ordered sets (not totally ordered) always contain a pair with intermediate probability of appearing in a certain order. This connects order theory with probability and has implications for sorting algorithms and social choice theory.
<!-- LITERATURE-TRIAGE:BEGIN -->... | open | L3: Advanced | 3 | Combinatorics | null | null | null | 124 | 9 | train |
NUM-003 | Infinitude of Perfect Numbers | Are there infinitely many perfect numbers? | All known perfect numbers are even and correspond to Mersenne primes via Euclid-Euler theorem. The question reduces to: are there infinitely many Mersenne primes? This remains open despite extensive computational searches. Connected to the distribution of primes and special number forms.
<!-- LITERATURE-TRIAGE:BEGIN -... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 345 | 27 | train |
NUM-005 | Lychrel Numbers | Do Lychrel numbers exist in base 10? | A Lychrel number never forms a palindrome through iterative reverse-and-add process. 196 is the first candidate—after billions of iterations, no palindrome found. Proving existence or non-existence would resolve this computational mystery connecting palindromes with iteration dynamics.
<!-- LITERATURE-TRIAGE:BEGIN -->... | open | L3: Advanced | 3 | Number Theory | null | null | null | 234 | 18 | train |
NUM-007 | Infinitude of Amicable Pairs | Are there infinitely many pairs of amicable numbers? | Amicable pairs (m,n) satisfy σ(m)-m=n and σ(n)-n=m. Over 12 million pairs known, but infinity unproven. Related to perfect numbers and sociable chains. Erdős-Rieger heuristics suggest infinity, but proof remains elusive.
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## Literature review (checked 2026-08-17)
**Status:** open
**C... | open | L4: Expert | 4 | Number Theory | null | null | null | 212 | 17 | train |
NUM-008 | Pi Normality | Is π a normal number (all digits equally frequent in all bases)? | A normal number has each digit appearing with equal asymptotic frequency in every base. While π appears statistically normal (verified to trillions of digits), no proof exists. This connects transcendental numbers, digit distribution, and randomness in mathematical constants.
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## Litera... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 389 | 30 | train |
NUM-009 | Algebraic Number Normality | Are all irrational algebraic numbers normal? | The question asks whether every irrational root of a polynomial with integer coefficients has all digits equally distributed in every base. A positive answer would be a remarkable connection between algebraic structure and digit statistics. Currently, we cannot prove normality for any specific algebraic irrational.
<!... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 201 | 16 | train |
NUM-013 | Hilbert's 12th Problem | Extend Kronecker-Weber theorem to abelian extensions of arbitrary number fields. | Hilbert's 12th problem asks for explicit construction of abelian extensions of number fields via special values of transcendental functions (generalizing cyclotomic fields for Q). Partial progress via complex multiplication, but general case remains one of Hilbert's unsolved problems.
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... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 187 | 15 | train |
NUM-015 | Siegel Zeros | Do Siegel zeros (real zeros of Dirichlet L-functions near s=1) exist? | Siegel zeros are hypothetical exceptional real zeros of L-functions very close to s=1. If they exist, they violate the Generalized Riemann Hypothesis. Their existence would have major consequences for prime distribution in arithmetic progressions. Most believe they don't exist.
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## Lite... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 234 | 18 | train |
NUM-016 | Schanuel's Conjecture | For e and π: are they algebraically independent? Is e+π, eπ, π^e, etc. transcendental? | Schanuel's conjecture is a fundamental statement about transcendence degrees. It implies e and π are algebraically independent and that expressions like e+π, eπ, π^π are transcendental. Proving it would resolve many open questions in transcendental number theory at once.
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## Literature ... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 287 | 22 | train |
NUM-017 | Euler-Mascheroni Constant Irrationality | Is the Euler-Mascheroni constant γ irrational? Transcendental? | The Euler-Mascheroni constant γ ≈ 0.5772 appears throughout analysis and number theory. We don't even know if it's irrational! Proving irrationality or transcendence would be a major achievement. Related constants like Catalan's G and ζ(3) face similar questions.
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## Literature review (... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 323 | 25 | train |
NUM-020 | Integer Factorization Polynomial Time | Can integer factorization be done in polynomial time? | The integer factorization problem asks whether factoring large integers into primes can be done efficiently (polynomial time). RSA cryptography relies on it being hard. Shor's algorithm solves it on quantum computers, but classical complexity remains unknown. Related to P vs NP.
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## Lit... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 456 | 35 | train |
PDE-001 | Navier-Stokes Existence and Smoothness | Do smooth solutions to Navier-Stokes equations exist globally in 3D? Or do finite-time singularities occur? | The Navier-Stokes existence and smoothness problem is one of the seven Millennium Prize Problems. It asks whether smooth solutions to the 3D Navier-Stokes equations exist for all time, or whether finite-time blow-up can occur. Fundamental for fluid dynamics and mathematical physics.
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##... | open | L5: Millennium Prize | 5 | Partial Differential Equations | null | null | null | 512 | 39 | train |
GEOM-001 | Sphere Packing Problem Higher Dimensions | What is the optimal sphere packing density in dimensions >3? | The sphere packing problem asks for the densest way to pack spheres in n-dimensional space. Solved in dimensions 1,2,3 (Kepler's conjecture, proved by Hales), 8, and 24 (Viazovska). Dimensions 4-7 and ≥9 remain open. Connections to lattices, coding theory, and optimization.
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## Literatu... | open | L5: Millennium Prize | 5 | Geometry | null | null | null | 298 | 23 | train |
HL-A | Hardy-Littlewood Conjecture A (Prime k-tuples) | Let $a_1, \ldots, a_k$ be given integers. Then there exist infinitely many positive integers $n$ such that $n + a_1, \ldots, n + a_k$ are all prime, provided that for every prime $p$, there exists an integer $m$ such that $(m + a_i, p) = 1$ for all $i$. | The first Hardy-Littlewood conjecture, also known as the prime k-tuples conjecture, generalizes the twin prime conjecture. It states that the asymptotic frequency of any admissible prime constellation can be computed explicitly. The case $k=2$ with $(a_1, a_2) = (0, 2)$ is the twin prime conjecture. Yitang Zhang proved... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 0 | 0 | train |
HL-B | Hardy-Littlewood Conjecture B (Second Conjecture) | For all integers $x, y \geq 2$, we have $\pi(x+y) \leq \pi(x) + \pi(y)$, where $\pi(n)$ denotes the prime counting function (the number of primes less than or equal to $n$). | The second Hardy-Littlewood conjecture states the subadditivity of the prime counting function. In 1974, Hensley and Richards proved that Conjecture A and Conjecture B are incompatible with each other - they cannot both be true. Since Conjecture A (the prime k-tuples conjecture) is considered more likely to be true bas... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 0 | 0 | train |
HL-F | Hardy-Littlewood Conjecture F (Primes in Quadratic Polynomials) | For a polynomial $f(x) = ax^2 + bx + c$ with $a > 0$, $\gcd(a,b,c) = 1$, and discriminant $\Delta = b^2 - 4ac$ not a perfect square, the polynomial takes infinitely many prime values. Furthermore, the number $P(n)$ of primes of the form $f(x) \leq n$ satisfies an asymptotic formula $P(n) \sim A \cdot \frac{\sqrt{n}}{\l... | Conjecture F is a special case of the Bateman-Horn conjecture and concerns primes represented by quadratic polynomials. It predicts not only the infinitude of such primes but also their asymptotic density. The constant A can take values larger or smaller than 1, meaning some polynomials are especially rich in primes wh... | open | L4: Expert | 4 | Number Theory | null | null | null | 0 | 0 | train |
GUY-A4 | The Prime Number Race | Let $\pi(n; a, b)$ be the number of primes $p \le n$ with $p \equiv a \pmod b$. For every $a$ and $b$ with $a \perp b$, are there infinitely many values of $n$ for which $\pi(n; a, b) > \pi(n; a_1, b)$ for every $a_1 \not\equiv a \pmod b$? | Turán was particularly interested in the prime number race. Knapowski & Turán settled special cases, but the general problem is wide open. Chebyshev noted that $\pi(n; 1, 3) < \pi(n; 2, 3)$ for small values of $n$, but this inequality is reversed for very large $n$. From Richard Guy's "Unsolved Problems in Number Theor... | open | L4: Expert | 4 | Number Theory | null | null | null | 0 | 0 | train |
GUY-A5b | Erdős $3000 Conjecture on Arithmetic Progressions | Let $\{a_i\}$ be any infinite sequence of integers for which $\sum 1/a_i$ is divergent. Does the sequence contain arbitrarily long arithmetic progressions? | Erdős offered $3000.00 for a proof or disproof of this conjecture. This is a generalization of the arithmetic progressions of primes problem. From Richard Guy's "Unsolved Problems in Number Theory", Section A5.
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## Literature review (checked 2026-08-17)
**Status:** partially_solved
*... | open | L4: Expert | 4 | Number Theory | null | null | null | 0 | 0 | train |
GUY-A6 | Consecutive Primes in Arithmetic Progression | Are there arbitrarily long arithmetic progressions of consecutive primes? That is, for any positive integer $k$, do there exist $k$ consecutive primes $p_n, p_{n+1}, \ldots, p_{n+k-1}$ in arithmetic progression? | Known examples include the 4-term sequences 251, 257, 263, 269 and 1741, 1747, 1753, 1759. Dubner, Forbes, Lygeros, Mizony & Zimmermann found 10 consecutive primes in arithmetic progression in 1998. It is not known if there are infinitely many sets of three consecutive primes in arithmetic progression. From Richard Guy... | open | L4: Expert | 4 | Number Theory | null | null | null | 0 | 0 | train |
GUY-A7a | Infinitude of Sophie Germain Primes | Are there infinitely many Sophie Germain primes? A prime $p$ is called a Sophie Germain prime if $2p + 1$ is also prime. | It is believed, but not known, that there are infinitely many Sophie Germain primes. Dubner has found many large examples. The largest known Sophie Germain prime has over 24000 decimal digits. From Richard Guy's "Unsolved Problems in Number Theory", Section A7.
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## Literature review (ch... | open | L4: Expert | 4 | Number Theory | null | null | null | 0 | 0 | train |
GUY-A7b | Shanks Chains of Length 7 | Are there any Shanks chains of length 7 with $p_{i+1} = 4p_i^2 - 17$? | Shanks chains are quadratic chains of primes. The recurrence $p_{i+1} = 4p_i^2 - 17$ yields a 4-chain if $p_1 = 3$ and a 5-chain if $p_1 = 303593$, but it can be seen (mod 59) that no such chain has length 17. It seems certain that such chains cannot be of arbitrary length. From Richard Guy's "Unsolved Problems in Numb... | open | L3: Advanced | 3 | Number Theory | null | null | null | 0 | 0 | train |
GUY-A8a | Erdős $5000 Problem on Prime Gaps | Is it true that for infinitely many $n$, $d_n = p_{n+1} - p_n > c \ln n \ln \ln n \ln \ln \ln \ln n / (\ln \ln \ln n)^2$ for arbitrarily large constant $c$? | Erdős offers $5,000 for a proof or disproof that the constant $c$ can be taken arbitrarily large. Rankin showed this holds for $c = e^\gamma$, and Pintz improved it to $c = 2e^\gamma > 3.562$. From Richard Guy's "Unsolved Problems in Number Theory", Section A8.
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## Literature review (ch... | open | L4: Expert | 4 | Number Theory | null | null | null | 0 | 0 | train |
GUY-A9 | General Patterns of Consecutive Primes | For any given pattern of primes with no congruence obstructions, are there infinitely many sets of consecutive primes with this pattern? | This conjecture is more general than Chowla's conjecture. It seems likely that there are infinitely many triples of primes $\{6k - 1, 6k + 1, 6k + 5\}$ and $\{6k + 1, 6k + 5, 6k + 7\}$. Hensley & Richards showed this is incompatible with the conjecture $\pi(x + y) \le \pi(x) + \pi(y)$ for all integers $x, y \ge 2$. Fro... | open | L4: Expert | 4 | Number Theory | null | null | null | 0 | 0 | train |
GUY-A11 | Erdős $100 Problem on Increasing and Decreasing Gaps | Does there exist an $n_0$ such that for every $i$ and $n > n_0$ we have $d_{n+2i} > d_{n+2i+1}$ and $d_{n+2i+1} < d_{n+2i+2}$, where $d_n = p_{n+1} - p_n$? | Erdős & Turán showed that the values of $n$ for which $d_n > d_{n+1}$ have positive lower density, but it is not known if there are infinitely many increasing or decreasing sets of three consecutive values of $d_n$. Erdős offers $100.00 for a proof that such an $n_0$ does not exist. From Richard Guy's "Unsolved Problem... | open | L3: Advanced | 3 | Number Theory | null | null | null | 0 | 0 | train |
GUY-A13 | Erdős Conjecture on Carmichael Numbers | Let $C(x)$ be the number of Carmichael numbers less than $x$. Does $(\ln C(x))/\ln x$ tend to 1 as $x$ tends to infinity? | Erdős conjectured this behavior for the count of Carmichael numbers. Alford, Granville & Pomerance showed there are infinitely many Carmichael numbers, in fact more than $x^\beta$ of them less than $x$ for $\beta > 0.290306$. Pomerance, Selfridge & Wagstaff give a heuristic argument supporting Erdős' conjecture. From R... | open | L4: Expert | 4 | Number Theory | null | null | null | 0 | 0 | train |
GUY-A14a | Pomerance's Questions on Good Primes | Call prime $p_n$ good if $p_n^2 > p_{n-i}p_{n+i}$ for all $i$, $1 \le i \le n-1$. Is it true that the set of $n$ for which $p_n$ is good has density 0? Are there infinitely many $n$ with $p_n p_{n+1} > p_{n-i} p_{n+1+i}$ for all $i$, $1 \le i \le n-1$? | Erdős and Straus introduced the concept of good primes. Examples include 5, 11, 17, and 29. Pomerance used the prime number graph to show there are infinitely many good primes and posed several related questions. From Richard Guy's "Unsolved Problems in Number Theory", Section A14.
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## ... | open | L3: Advanced | 3 | Number Theory | null | null | null | 0 | 0 | train |
GUY-A15 | Congruent Products of Consecutive Numbers | What is the least prime $p$ such that there are integers $a, k_1, k_2, k_3$ with $\prod_{i=1}^{k_1} (a+i) \equiv \prod_{i=1}^{k_2} (a+k_1+i) \equiv \prod_{i=1}^{k_3} (a+k_1+k_2+i) \equiv 1 \pmod{p}$? | Erdős observed that $3 \cdot 4 \equiv 5 \cdot 6 \cdot 7 \equiv 1 \pmod{11}$ and suggested that such primes $p$ exist for any number of congruent products. Narkiewicz and others found examples for larger numbers of terms. From Richard Guy's "Unsolved Problems in Number Theory", Section A15.
<!-- LITERATURE-TRIAGE:BEGIN... | open | L2: Intermediate | 2 | Number Theory | null | null | null | 0 | 0 | train |
GUY-A16 | Walking to Infinity on Gaussian Primes | Can one walk from the origin to infinity using Gaussian primes as stepping stones and taking steps of bounded length? | Motzkin and Gordon asked this question about Gaussian primes (primes in the ring of complex numbers $a+bi$ where $a, b$ are integers). Presumably not. Jordan & Rabung showed that steps of length at least 4 are necessary. Gethner, Wagon & Wick produced a moat of width $\sqrt{26}$. From Richard Guy's "Unsolved Problems i... | open | L3: Advanced | 3 | Number Theory | null | null | null | 0 | 0 | train |
GUY-A17 | Giuga's Conjecture on Prime Characterization | Is it true that if $n$ divides $1^{n-1} + 2^{n-1} + \dots + (n-1)^{n-1} + 1$, then $n$ is prime? | Sierpiński observed that if $n$ is prime, then $n$ divides this sum. Giuga conjectured the converse and verified it for $n \le 10^{1000}$. A counterexample would be a Carmichael number with additional properties. An equivalent conjecture is $n B_{n-1} \equiv -1 \pmod{n}$ where $B_k$ are Bernoulli numbers. From Richard ... | open | L3: Advanced | 3 | Number Theory | null | null | null | 0 | 0 | train |
GUY-A18 | Erdős-Selfridge Classification: Infinitely Many Primes in Each Class | In the Erdős-Selfridge classification of primes, are there infinitely many primes in each class? Prime $p$ is in class 1 if the only prime divisors of $p+1$ are 2 or 3; and $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\le r-1$, with equality for at least one prime factor. | The first few classes are: Class 1: 2, 3, 5, 7, 11, 17, 23, 31, 47, 53, 71, 107, 127, 191, ...; Class 2: 13, 19, 29, 41, 43, 59, 61, 67, 79, 83, 89, 97, 101, ...; Class 3: 37, 103, 113, 151, 157, 163, 173, 181, 193, 227, 233, ... From Richard Guy's "Unsolved Problems in Number Theory", Section A18.
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GUY-A19a | Erdős Conjecture on $n - 2^k$ Prime | Are 4, 7, 15, 21, 45, 75, and 105 the only values of $n$ for which $n - 2^k$ is prime for all $k$ such that $2 \le 2^k < n$? | Erdős conjectures that these are the only such values. He also conjectures that for infinitely many $n$, all the integers $n - 2^k, 1 \le 2^k < n$ are squarefree. From Richard Guy's "Unsolved Problems in Number Theory", Section A19.
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## Literature review (checked 2026-08-17)
**Status:*... | open | L3: Advanced | 3 | Number Theory | null | null | null | 0 | 0 | train |
GUY-A19b | Cohen-Selfridge Problem on $\pm p^a \pm 2^b$ | What is the least positive odd number not of the form $\pm p^a \pm 2^b$, where $p$ is an odd prime? | Cohen & Selfridge observed that the number is greater than $2^{18}$. This is related to the representation of odd numbers as sums or differences of prime powers and powers of 2. From Richard Guy's "Unsolved Problems in Number Theory", Section A19.
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## Literature review (checked 2026-08-... | open | L2: Intermediate | 2 | Number Theory | null | null | null | 0 | 0 | train |
GUY-A20 | Density of Symmetric Primes | Given pairs of odd primes $p, q$, define $S(q,p)$ as the number of lattice points $(m, n)$ in the rectangle $0 < m < p/2$, $0 < n < q/2$ below the diagonal. A pair is symmetric if $S(p, q) = S(q,p)$. Is the number of symmetric primes less than $x$ equal to $x/(\ln x)^{\sigma+o(1)}$, where $\sigma = 2 - (1+\ln \ln 2)/\l... | Fletcher, Lindgren & Pomerance showed that a pair is symmetric just if $|p - q| = (p - 1, q - 1)$, and that the number of symmetric primes less than $x$ is at most $x/(\ln x)^{1.027}$. They conjectured the more precise asymptotic. From Richard Guy's "Unsolved Problems in Number Theory", Section A20.
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GUY-A12a | Square Pseudoprimes | Are there any square pseudoprimes (base 2) other than multiples of $1194649 = 1093^2$ or $12327121 = 3511^2$? | Pinch observed that there are 54 non-squarefree pseudoprimes up to $10^{13}$, all multiples of $1093^2$ or $3511^2$. The question asks if there are other perfect squares that are pseudoprimes to base 2. From Richard Guy's "Unsolved Problems in Number Theory", Section A12.
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## Literature... | open | L3: Advanced | 3 | Number Theory | null | null | null | 0 | 0 | train |
GUY-A12b | Selfridge-Wagstaff-Pomerance Prize Problem | Does there exist a composite number $n \equiv 3$ or $7 \pmod{10}$ which divides both $2^n - 2$ and the Fibonacci number $u_{n+1}$? | Selfridge, Wagstaff & Pomerance offer $500 + $100 + $20 = $620 for finding such a composite $n$, or $20 + $100 + $500 = $620 for a proof that no such $n$ exists. This combines pseudoprime properties with Fibonacci divisibility. From Richard Guy's "Unsolved Problems in Number Theory", Section A12.
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GUY-A12c | Even Fibonacci Pseudoprimes | Does there exist an even Fibonacci pseudoprime? | A Fibonacci pseudoprime of the $m$-th kind is an odd composite integer $n$ with $V_n(m, -1) \equiv m \pmod n$ where $V_n$ is the Lucas sequence. Somer showed that if an even Fibonacci pseudoprime exists, it must be greater than $28 \times 10^{12}$. From Richard Guy's "Unsolved Problems in Number Theory", Section A12.
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EP-3 | Erdős Problem #3 | If $A\subseteq \mathbb{N}$ has $\sum_{n\in A}\frac{1}{n}=\infty$ then must $A$ contain arbitrarily long arithmetic progressions? | This is essentially asking for good bounds on $r_k(N)$, the size of the largest subset of $\{1,\ldots,N\}$ without a non-trivial $k$-term arithmetic progression. For example, a bound like $ r_k(N) \ll_k \frac{N}{(\log N)(\log\log N)^2} $ would be sufficient.
Even the case $k=3$ is non-trivial, but was proved by Bloom a... | open | L1: Tractable | 1 | Number Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-5 | Erdős Problem #5 | Let $C\geq 0$. Is there an infinite sequence of $n_i$ such that $ \lim_{i\to \infty}\frac{p_{n_i+1}-p_{n_i}}{\log n_i}=C? $ | Let $S$ be the set of limit points of $(p_{n+1}-p_n)/\log n$. This problem asks whether $S=[0,\infty]$. Although this conjecture remains unproven, a lot is known about $S$. Some highlights:
{UL}
{LI}$\infty\in S$ by Westzynthius' result \cite{We31} on large prime gaps,{/LI}
{LI}$0\in S$ by the work of Goldston, Pintz, ... | open | L1: Tractable | 1 | Number Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-9 | Erdős Problem #9 | Let $A$ be the set of all odd integers not of the form $p+2^{k}+2^l$ (where $k,l\geq 0$ and $p$ is prime). Is the upper density of $A$ positive? | In \cite{Er77c} Erd\H{o}s credits Schinzel with proving that there are infinitely many odd integers not of this form, but gives no reference. Crocker \cite{Cr71} has proved there are $\gg\log\log N$ such integers in $\{1,\ldots,N\}$. Pan \cite{Pa11} improved this to $\gg_\epsilon N^{1-\epsilon}$ for any $\epsilon>0$. E... | open | L1: Tractable | 1 | Number Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-10 | Erdős Problem #10 | Is there some $k$ such that every integer is the sum of a prime and at most $k$ powers of 2? | Erd\H{o}s described this as 'probably unattackable'. In \cite{ErGr80} Erd\H{o}s and Graham suggest that no such $k$ exists. Gallagher \cite{Ga75} has shown that for any $\epsilon>0$ there exists $k(\epsilon)$ such that the set of integers which are the sum of a prime and at most $k(\epsilon)$ many powers of 2 has lower... | open | L1: Tractable | 1 | Number Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-12 | Erdős Problem #12 | Let $A$ be an infinite set such that there are no distinct $a,b,c\in A$ such that $a\mid (b+c)$ and $b,c>a$. Is there such an $A$ with $ \liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}>0? $ Does there exist some absolute constant $c>0$ such that there are always infinitely many $N$ with $ \lvert A\cap\{1,\ldot... | Asked by Erd\H{o}s and S\'{a}rk"{o}zy \cite{ErSa70}, who proved that $A$ must have density $0$. They also prove that this is essentially best possible, in that given any function $f(x)\to \infty$ as $x\to \infty$ there exists a set $A$ with this property and infinitely many $N$ such that $ \lvert A\cap\{1,\ldots,N\}\rv... | open | L1: Tractable | 1 | Number Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-14 | Erdős Problem #14 | Let $A\subseteq \mathbb{N}$. Let $B\subseteq \mathbb{N}$ be the set of integers which are representable in exactly one way as the sum of two elements from $A$.
Is it true that for all $\epsilon>0$ and large $N$ $ \lvert \{1,\ldots,N\}\backslash B\rvert \gg_\epsilon N^{1/2-\epsilon}? $ Is it possible that $ \lvert \{1,\... | Apparently originally considered by Erd\H{o}s and Nathanson, although later Erd\H{o}s attributes this to Erd\H{o}s, S\'{a}rk"{o}zy, and Szemer\'{e}di (but gives no reference), and claims a construction of an $A$ such that for all $\epsilon>0$ and all large $N$ $ \lvert \{1,\ldots,N\}\backslash B\rvert \ll_\epsilon N^{1... | open | L1: Tractable | 1 | Number Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-15 | Erdős Problem #15 | Is it true that $ \sum_{n=1}^\infty(-1)^n\frac{n}{p_n} $ converges, where $p_n$ is the sequence of primes? | Erd\H{o}s suggested that a computer could be used to explore this, and did not see any other method to attack this.
Tao \cite{Ta23} has proved that this series does converge assuming a strong form of the Hardy-Littlewood prime tuples conjecture.
In \cite{Er98} Erd\H{o}s further conjectures that $ \sum_{n=1}^\infty (-1)... | open | L1: Tractable | 1 | Number Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-17 | Erdős Problem #17 | Are there infinitely many primes $p$ such that every even number $n\leq p-3$ can be written as a difference of primes $n=q_1-q_2$ where $q_1,q_2\leq p$? | The first prime without this property is $97$. The sequence of such primes is A038133 in the OEIS. These are called cluster primes.
Blecksmith, Erd\H{o}s, and Selfridge \cite{BES99} proved that the number of such primes is $ \ll_A \frac{x}{(\log x)^A} $ for every $A>0$, and Elsholtz \cite{El03} improved this to $ \ll x... | open | L1: Tractable | 1 | Number Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-18 | Erdős Problem #18 | We call $m$ practical if every integer $n<m$ is the sum of distinct divisors of $m$. If $m$ is practical then let $h(m)$ be such that $h(m)$ many divisors always suffice.
Are there infinitely many practical $m$ such that $ h(m) < (\log\log m)^{O(1)}? $ Is it true that $h(n!)<n^{o(1)}$? Or perhaps even $h(n!)<(\log n)^{... | It is easy to see that almost all numbers are not practical. Erd\H{o}s originally showed that $h(n!) <n$. Vose \cite{Vo85} proved the existence of infinitely many practical $m$ such that $h(m)\ll (\log m)^{1/2}$.
The sequence of practical numbers is A005153 in the OEIS.
The reward of \$250 is offered in \cite{Er81h}, a... | open | L1: Tractable | 1 | Number Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-25 | Erdős Problem #25 | Let $n_1<n_2<\cdots$ be an arbitrary sequence of integers, each with an associated residue class $a_i\pmod{n_i}$. Let $A$ be the set of integers $n$ such that for every $i$ either $n<n_i$ or $n
ot\equiv a_i\pmod{n_i}$. Must the logarithmic density of $A$ exist? | This is a special case of [486].
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** The one-residue-class-per-modulus logarithmic-density question remains open; it is a special case of EP-486.
**Verified pa... | open | L1: Tractable | 1 | Number Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-28 | Erdős Problem #28 | If $A\subseteq \mathbb{N}$ is such that $A+A$ contains all but finitely many integers then $\limsup 1_A\ast 1_A(n)=\infty$. | Conjectured by Erd\H{o}s and Tur\'{a}n. They also suggest the stronger conjecture that $\limsup 1_A\ast 1_A(n)/\log n>0$.
Another stronger conjecture would be that the hypothesis $\lvert A\cap [1,N]\rvert \gg N^{1/2}$ for all large $N$ suffices.
Erd\H{o}s and S\'{a}rk"{o}zy conjectured the stronger version that if $A=\... | open | L1: Tractable | 1 | Number Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-30 | Erdős Problem #30 | Let $h(N)$ be the maximum size of a Sidon set in $\{1,\ldots,N\}$. Is it true that, for every $\epsilon>0$, $ h(N) = N^{1/2}+O_\epsilon(N^\epsilon)? $ | A problem of Erd\H{o}s and Tur\'{a}n. It may even be true that $h(N)=N^{1/2}+O(1)$, but Erd\H{o}s remarks this is perhaps too optimistic. Erd\H{o}s and Tur\'{a}n \cite{ErTu41} proved an upper bound of $N^{1/2}+O(N^{1/4})$, with an alternative proof by Lindstr"{o}m \cite{Li69}. Both proofs in fact give $ h(N) \leq N^{1/... | open | L1: Tractable | 1 | Number Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-32 | Erdős Problem #32 | Is there a set $A\subset\mathbb{N}$ such that $ \lvert A\cap\{1,\ldots,N\}\rvert = o((\log N)^2) $ and such that every large integer can be written as $p+a$ for some prime $p$ and $a\in A$?
Can the bound $O(\log N)$ be achieved? Must such an $A$ satisfy $ \liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{\log N}> 1? $ | Such a set is called an additive complement to the primes.
Erd\H{o}s \cite{Er54} proved that such a set $A$ exists with $\lvert A\cap\{1,\ldots,N\}\rvert\ll (\log N)^2$ (improving a previous result of Lorentz \cite{Lo54} who achieved $\ll (\log N)^3$).
Wolke \cite{Wo96} has shown that such a bound is almost true, in th... | open | L1: Tractable | 1 | Number Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-33 | Erdős Problem #33 | Let $A\subset\mathbb{N}$ be such that every large integer can be written as $n^2+a$ for some $a\in A$ and $n\geq 0$. What is the smallest possible value of $ \limsup \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}? $ Is $ \liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}>1? $ | Such a set $A$ is called an additive complement of the set of squares. Erd\H{o}s observed that there exist $A$ for which the $\limsup$ is finite and $>1$. Moser \cite{Mo65} proved that, for any such $A$, $ \liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}>1.06. $ The best-known lower bound is $ \liminf \frac{\lv... | open | L1: Tractable | 1 | Number Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-36 | Erdős Problem #36 | Find the optimal constant $c>0$ such that the following holds.
For all sufficiently large $N$, if $A\sqcup B=\{1,\ldots,2N\}$ is a partition into two equal parts, so that $\lvert A\rvert=\lvert B\rvert=N$, then there is some $x$ such that the number of solutions to $a-b=x$ with $a\in A$ and $b\in B$ is at least $cN$. | The minimum overlap problem. The example (with $N$ even) $A=\{N/2+1,\ldots,3N/2\}$ shows that $c\leq 1/2$ (indeed, Erd\H{o}s initially conjectured that $c=1/2$). The lower bound of $c\geq 1/4$ is trivial, and Scherk improved this to $1-1/\sqrt{2}=0.29\cdots$. The current records are $ 0.379005 < c < 0.380924, $ the low... | open | L1: Tractable | 1 | Number Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-39 | Erdős Problem #39 | Is there an infinite Sidon set $A\subset \mathbb{N}$ such that $ \lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon} $ for all $\epsilon>0$? | The trivial greedy construction achieves $\gg N^{1/3}$. The first improvement on this was achieved by Ajtai, Koml\'{o}s, and Szemer\'{e}di \cite{AKS81b}, who found an infinite Sidon set with growth rate $\gg (N\log N)^{1/3}$. The current best bound of $\gg N^{\sqrt{2}-1+o(1)}$ is due to Ruzsa \cite{Ru98}.
Erd\H{o}s \ci... | open | L1: Tractable | 1 | Combinatorics | Erdős Problems | null | null | 0 | 0 | train |
EP-40 | Erdős Problem #40 | For what functions $g(N)\to \infty$ is it true that $ \lvert A\cap \{1,\ldots,N\}\rvert \gg \frac{N^{1/2}}{g(N)} $ implies $\limsup 1_A\ast 1_A(n)=\infty$? | This is a stronger form of the Erd\H{o}s-Tur\'{a}n conjecture [28] (since establishing this for any function $g(N)\to \infty$ would imply a positive solution to [28]).
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature as... | open | L1: Tractable | 1 | Combinatorics | Erdős Problems | null | null | 0 | 0 | train |
EP-41 | Erdős Problem #41 | Let $A\subset\mathbb{N}$ be an infinite set such that the triple sums $a+b+c$ are all distinct for $a,b,c\in A$ (aside from the trivial coincidences). Is it true that $ \liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/3}}=0? $ | Erd\H{o}s proved that if the pairwise sums $a+b$ are all distinct aside from the trivial coincidences then $ \liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}=0. $ This is discussed in problem C11 of Guy's collection \cite{Gu04}, in which Guy says Erd\H{o}s offered \$500 for the general problem of whether, for ... | open | L1: Tractable | 1 | Combinatorics | Erdős Problems | null | null | 0 | 0 | train |
EP-42 | Erdős Problem #42 | Let $M\geq 1$ and $N$ be sufficiently large in terms of $M$. Is it true that for every Sidon set $A\subset \{1,\ldots,N\}$ there is another Sidon set $B\subset \{1,\ldots,N\}$ of size $M$ such that $(A-A)\cap(B-B)=\{0\}$? | <!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** solved
**Classification:** SOLVED-IN-LITERATURE
**Current literature assessment.** The maintained tracker records SOLVED (LEAN). A pinned Lean development proves the all-M eventual existence theorem, and a separate bridge extract... | open | L1: Tractable | 1 | Combinatorics | Erdős Problems | null | null | 0 | 0 | train |
EP-43 | Erdős Problem #43 | If $A,B\subset \{1,\ldots,N\}$ are two Sidon sets such that $(A-A)\cap(B-B)=\{0\}$ then is it true that $ \binom{\lvert A\rvert}{2}+\binom{\lvert B\rvert}{2}\leq\binom{f(N)}{2}+O(1), $ where $f(N)$ is the maximum possible size of a Sidon set in $\{1,\ldots,N\}$? If $\lvert A\rvert=\lvert B\rvert$ then can this bound b... | Since it is known that $f(N)\sim \sqrt{N}$ (see [30]) the latter question is equivalent to asking whether, if $\lvert A\rvert=\lvert B\rvert$, $ \lvert A\rvert \leq \left(\frac{1}{\sqrt{2}}-c+o(1)\right)\sqrt{N} $ for some constant $c>0$. In the comments Tao has given a proof of this upper bound without the $-c$.
In th... | open | L1: Tractable | 1 | Combinatorics | Erdős Problems | null | null | 0 | 0 | train |
EP-44 | Erdős Problem #44 | Let $N\geq 1$ and $A\subset \{1,\ldots,N\}$ be a Sidon set. Is it true that, for any $\epsilon>0$, there exist $M$ and $B\subset \{N+1,\ldots,M\}$ (which may depend on $N,A,\epsilon$) such that $A\cup B\subset \{1,\ldots,M\}$ is a Sidon set of size at least $(1-\epsilon)M^{1/2}$? | See also [329] and [707] (indeed a positive solution to [707] implies a positive solution to this problem, which in turn implies a positive solution to [329]).
This is discussed in problem C9 of Guy's collection \cite{Gu04}.
References
[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.
<!... | open | L1: Tractable | 1 | Combinatorics | Erdős Problems | null | null | 0 | 0 | train |
EP-50 | Erdős Problem #50 | Schoenberg proved that for every $c\in [0,1]$ the density of $ \{ n\in \mathbb{N} : \phi(n)<cn\} $ exists. Let this density be denoted by $f(c)$. Is it true that there are no $x$ such that $f'(x)$ exists and is positive? | Erd\H{o}s \cite{Er95} could prove the distribution function is purely singular.
References
[Er95] Erd\H{o}s, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Cl... | open | L1: Tractable | 1 | Combinatorics | Erdős Problems | null | null | 0 | 0 | train |
EP-51 | Erdős Problem #51 | Is there an infinite set $A\subset \mathbb{N}$ such that for every $a\in A$ there is an integer $n$ such that $\phi(n)=a$, and yet if $n_a$ is the smallest such integer then $n_a/a\to \infty$ as $a\to\infty$? | Carmichael has asked whether there is an integer $t$ for which $\phi(n)=t$ has exactly one solution. Erd\H{o}s has proved that if such a $t$ exists then there must be infinitely many such $t$.
See also [694].
This is discussed in problems B36 and B39 of Guy's collection \cite{Gu04}.
References
[Gu04] Guy, Richard K.,... | open | L1: Tractable | 1 | Number Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-52 | Erdős Problem #52 | Let $A$ be a finite set of integers. Is it true that for every $\epsilon>0$ $ \max( \lvert A+A\rvert,\lvert AA\rvert)\gg_\epsilon \lvert A\rvert^{2-\epsilon}? $ | The sum-product problem. Erd\H{o}s and Szemer\'{e}di \cite{ErSz83} proved a lower bound of $\lvert A\rvert^{1+c}$ for some constant $c>0$, and an upper bound of $ \lvert A\rvert^2 \exp\left(-c\frac{\log\lvert A\rvert}{\log\log \lvert A\rvert}\right) $ for some constant $c>0$. The lower bound has been improved a number ... | open | L1: Tractable | 1 | Number Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-60 | Erdős Problem #60 | Does every graph on $n$ vertices with $>\mathrm{ex}(n;C_4)$ edges contain $\gg n^{1/2}$ many copies of $C_4$? | Conjectured by Erd\H{o}s and Simonovits, who could not even prove that at least $2$ copies of $C_4$ are guaranteed.
The behaviour of $\mathrm{ex}(n;C_4)$ is the subject of [765].
He, Ma, and Yang \cite{HeMaYa21} have proved this conjecture when $n=q^2+q+1$ for some even integer $q$.
References
[HeMaYa21] He, J. and M... | open | L1: Tractable | 1 | Number Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-61 | Erdős Problem #61 | For any graph $H$ is there some $c=c(H)>0$ such that every graph $G$ on $n$ vertices that does not contain $H$ as an induced subgraph contains either a complete graph or independent set on $\geq n^c$ vertices? | Conjectured by Erd\H{o}s and Hajnal \cite{ErHa89}, who proved that a complete graph or independent set must exist on $ \geq \exp(c_H\sqrt{\log n}) $ many vertices, where $c_H>0$ is some constant. This was improved by Buci\'{c}, Nguyen, Scott, and Seymour \cite{BNSS23} to $ \geq \exp(c_H\sqrt{\log n\log\log n}). $ See a... | open | L1: Tractable | 1 | Graph Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-62 | Erdős Problem #62 | If $G_1,G_2$ are two graphs with chromatic number $\aleph_1$ then must there exist a graph $G$ whose chromatic number is $4$ (or even $\aleph_0$) which is a subgraph of both $G_1$ and $G_2$? | Erd\H{o}s also asked \cite{Er87} about finding a common subgraph $H$ (with chromatic number either $4$ or $\aleph_0$) in any finite collection of graphs with chromatic number $\aleph_1$.
Every graph with chromatic number $\aleph_1$ contains all sufficiently large odd cycles (which have chromatic number $3$), see [594].... | open | L1: Tractable | 1 | Graph Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-65 | Erdős Problem #65 | Let $G$ be a graph with $n$ vertices and $kn$ edges, and $a_1<a_2<\cdots $ be the lengths of cycles in $G$. Is it true that $ \sum\frac{1}{a_i}\gg \log k? $ Is the sum $\sum\frac{1}{a_i}$ minimised when $G$ is a complete bipartite graph? | A problem of Erd\H{o}s and Hajnal.
Gy\'{a}rf\'{a}s, Koml\'{o}s, and Szemer\'{e}di \cite{GKS84} have proved that this sum is $\gg \log k$, so that only the second question remains. Liu and Montgomery \cite{LiMo20} have proved the asymptotically sharp lower bound of $\geq (\tfrac{1}{2}-o(1))\log k$.
See also the entry in... | open | L1: Tractable | 1 | Graph Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-66 | Erdős Problem #66 | Is there $A\subseteq \mathbb{N}$ such that $ \lim_{n\to \infty}\frac{1_A\ast 1_A(n)}{\log n} $ exists and is $
eq 0$? | A suitably constructed random set has this property if we are allowed to ignore an exceptional set of density zero. The challenge is obtaining this with no exceptional set. Erd\H{o}s believed the answer should be no. Erd\H{o}s and S\'{a}rk"{o}zy proved that $ \frac{\lvert 1_A\ast 1_A(n)-\log n\rvert}{\sqrt{\log n}}\to ... | open | L1: Tractable | 1 | Combinatorics | Erdős Problems | null | null | 0 | 0 | train |
EP-68 | Erdős Problem #68 | Is $ \sum_{n\geq 2}\frac{1}{n!-1} $ irrational? | The decimal expansion is A331373 in the OEIS. Weisenberg has observed that this sum can also be written as $ \sum_{k\geq 1}\sum_{n\geq 2}\frac{1}{(n!)^k}. $ Erd\H{o}s \cite{Er88c} notes that $\sum \frac{1}{n!+t}$ should be transcendental for every integer $t$.
References
[Er88c] Erd"{o}s, P., On the irrationality of ... | open | L1: Tractable | 1 | Number Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-70 | Erdős Problem #70 | Let $\mathfrak{c}$ be the ordinal of the real numbers, $\beta$ be any countable ordinal, and $2\leq n<\omega$. Is it true that $\mathfrak{c}\to (\beta, n)_2^3$? | Erd\H{o}s and Rado proved that $\mathfrak{c}\to (\omega+n,4)_2^3$ for any $2\leq n<\omega$.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** The full triple partition relation for arbitrary countable beta ... | open | L1: Tractable | 1 | Set Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-74 | Erdős Problem #74 | Let $f(n)\to \infty$ (possibly very slowly). Is there a graph of infinite chromatic number such that every finite subgraph on $n$ vertices can be made bipartite by deleting at most $f(n)$ edges? | Conjectured by Erd\H{o}s, Hajnal, and Szemer\'{e}di \cite{EHS82}.
R"{o}dl \cite{Ro82} has proved this for hypergraphs, and also proved there is such a graph (with chromatic number $\aleph_0$) if $f(n)=\epsilon n$ for any fixed constant $\epsilon>0$.
It is open even for $f(n)=\sqrt{n}$. Erd\H{o}s offered \$500 for a pro... | open | L1: Tractable | 1 | Graph Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-75 | Erdős Problem #75 | Is there a graph of chromatic number $\aleph_1$ such that for all $\epsilon>0$ if $n$ is sufficiently large and $H$ is a subgraph on $n$ vertices then $H$ contains an independent set of size $>n^{1-\epsilon}$? | Conjectured by Erd\H{o}s, Hajnal, and Szemer\'{e}di \cite{EHS82}. In \cite{Er95d} Erd\H{o}s suggests this may even be true with an independent set of size $\gg n$.
See also [750].
References
[EHS82] Erd\H{o}s, P. and Hajnal, A. and Szemer\'{e}di, E., On almost bipartite large chromatic graphs. Theory and practice of ... | open | L1: Tractable | 1 | Graph Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-77 | Erdős Problem #77 | If $R(k)$ is the Ramsey number for $K_k$, the minimal $n$ such that every $2$-colouring of the edges of $K_n$ contains a monochromatic copy of $K_k$, then find the value of $ \lim_{k\to \infty}R(k)^{1/k}. $ | Erd\H{o}s offered \$100 for just a proof of the existence of this constant, without determining its value. He also offered \$1000 for a proof that the limit does not exist, but says 'this is really a joke as [it] certainly exists'. (In \cite{Er88} he raises this prize to \$10000). Erd\H{o}s proved $ \sqrt{2}\leq \limin... | open | L1: Tractable | 1 | Graph Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-78 | Erdős Problem #78 | Give a constructive proof that $R(k)>C^k$ for some constant $C>1$. | Erd\H{o}s gave a simple probabilistic proof that $R(k) \gg k2^{k/2}$.
Equivalently, this question asks for an explicit construction of a graph on $n$ vertices which does not contain any clique or independent set of size $\geq c\log n$ for some constant $c>0$.
In \cite{Er69b} Erd\H{o}s asks for even a construction whose... | open | L1: Tractable | 1 | Graph Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-82 | Erdős Problem #82 | Let $F(n)$ be maximal such that every graph on $n$ vertices contains a regular induced subgraph on at least $F(n)$ vertices. Prove that $F(n)/\log n\to \infty$. | Conjectured by Erd\H{o}s, Fajtlowicz, and Stanton. It is known that $F(5)=3$ and $F(7)=4$.
Ramsey's theorem implies that $F(n)\gg \log n$. Bollob\'{a}s observed that $F(n)\ll n^{1/2+o(1)}$. Alon, Krivelevich, and Sudakov \cite{AKS07} have improved this to $n^{1/2}(\log n)^{O(1)}$.
In \cite{Er93} Erd\H{o}s asks whether,... | open | L1: Tractable | 1 | Graph Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-84 | Erdős Problem #84 | The cycle set of a graph $G$ on $n$ vertices is a set $A\subseteq \{3,\ldots,n\}$ such that there is a cycle in $G$ of length $\ell$ if and only if $\ell \in A$. Let $f(n)$ count the number of possible such $A$.
Prove that $f(n)=o(2^n)$.
Prove that $f(n)/2^{n/2}\to \infty$. | Conjectured by Erd\H{o}s and Faudree, who showed that $2^{n/2}<f(n) \leq 2^{n-2}$. The first problem was solved by Verstra"{e}te \cite{Ve04}, who proved $ f(n)\ll 2^{n-n^{1/10}}. $ This was improved by Nenadov \cite{Ne25} to $ f(n) \ll 2^{n-n^{1/2-o(1)}}. $ One can also ask about the existence and value of $\lim f(n)^{... | open | L1: Tractable | 1 | Graph Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-86 | Erdős Problem #86 | Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Is it true that every subgraph of $Q_n$ with $ \geq \left(\frac{1}{2}+o(1)\right)n2^{n-1} $ many edges contains a $C_4$? | Let $f(n)$ be the maximum number of edges in a subgraph of $Q_n$ without a $C_4$, so that this conjecture is that $f(n)\leq (\frac{1}{2}+o(1))n2^{n-1}$.
Erd\H{o}s \cite{Er91} showed that $ f(n) \geq \left(\frac{1}{2}+\frac{c}{n}\right)n2^{n-1} $ for some constant $c>0$, and wrote it is 'perhaps not hopeless' to determi... | open | L1: Tractable | 1 | Graph Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-89 | Erdős Problem #89 | Does every set of $n$ distinct points in $\mathbb{R}^2$ determine $\gg n/\sqrt{\log n}$ many distinct distances? | A $\sqrt{n}\times\sqrt{n}$ integer grid shows that this would be the best possible. Nearly solved by Guth and Katz \cite{GuKa15} who proved that there are always $\gg n/\log n$ many distinct distances.
A stronger form (see [604]) may be true: is there a single point which determines $\gg n/\sqrt{\log n}$ distinct dista... | open | L1: Tractable | 1 | Number Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-90 | Erdős Problem #90 | Does every set of $n$ distinct points in $\mathbb{R}^2$ contain at most $n^{1+O(1/\log\log n)}$ many pairs which are distance 1 apart? | The unit distance problem. In \cite{Er94b} Erd\H{o}s dates this conjecture to 1946. In \cite{Er82e} he offers \$300 for the upper bound $n^{1+o(1)}$.
This would be the best possible, as is shown by a set of lattice points. It is easy to show that there are $O(n^{3/2})$ many such pairs. The best known upper bound is $O(... | open | L1: Tractable | 1 | Graph Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-91 | Erdős Problem #91 | Let $n$ be a sufficently large integer. Suppose $A\subset \mathbb{R}^2$ has $\lvert A\rvert=n$ and minimises the number of distinct distances between points in $A$. Prove that there are at least two (and probably many) such $A$ which are non-similar. | For $n=3$ the equilateral triangle is the only such set. For $n=4$ the square or two equilateral triangles sharing an edge give two non-similar examples.
For $n=5$ the regular pentagon is the unique such set (which has two distinct distances). Erd\H{o}s mysteriously remarks in \cite{Er90} this was proved by 'a colleagu... | open | L1: Tractable | 1 | Number Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-92 | Erdős Problem #92 | Let $f(n)$ be maximal such that there exists a set $A$ of $n$ points in $\mathbb{R}^2$ in which every $x\in A$ has at least $f(n)$ points in $A$ equidistant from $x$.
Is it true that $f(n)\leq n^{o(1)}$? Or even $f(n) < n^{O(1/\log\log n)}$? | This is a stronger form of the unit distance conjecture (see [90]).
The set of lattice points imply $f(n) > n^{c/\log\log n}$ for some constant $c>0$. Erd\H{o}s offered \$500 for a proof that $f(n) \leq n^{o(1)}$ but only \$100 for a counterexample. This latter prize is downgraded to \$50 in \cite{ErFi97}.
It is trivia... | open | L1: Tractable | 1 | Graph Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-96 | Erdős Problem #96 | If $n$ points in $\mathbb{R}^2$ form a convex polygon then there are $O(n)$ many pairs which are distance $1$ apart. | Conjectured by Erd\H{o}s and Moser. In \cite{Er92e} Erd\H{o}s credits the conjecture that the true upper bound is $2n$ to himself and Fishburn. F"{u}redi \cite{Fu90} proved an upper bound of $O(n\log n)$. A short proof of this bound was given by Brass and Pach \cite{BrPa01}. The best known upper bound is $ \leq n\log_2... | open | L1: Tractable | 1 | Graph Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-98 | Erdős Problem #98 | Let $h(n)$ be such that any $n$ points in $\mathbb{R}^2$, with no three on a line and no four on a circle, determine at least $h(n)$ distinct distances. Does $h(n)/n\to \infty$? | Erd\H{o}s could not even prove $h(n)\geq n$. Pach has shown $h(n)<n^{\log_23}$. Erd\H{o}s, F"{u}redi, and Pach \cite{EFPR93} have improved this to $ h(n) < n\exp(c\sqrt{\log n}) $ for some constant $c>0$.
References
[EFPR93] Erd\H{o}s, Paul and F"{u}redi, Zolt\'{a}n and Pach, J\'{a}nos and
Ruzsa, Imre Z., The grid re... | open | L1: Tractable | 1 | Geometry | Erdős Problems | null | null | 0 | 0 | train |
EP-99 | Erdős Problem #99 | Let $A\subseteq\mathbb{R}^2$ be a set of $n$ points with minimum distance equal to 1, chosen to minimise the diameter of $A$. If $n$ is sufficiently large then must there be three points in $A$ which form an equilateral triangle of size 1? | Thue proved that the minimal such diameter is achieved (asymptotically) by the points in a triangular lattice intersected with a circle. In general Erd\H{o}s believed such a set must have very large intersection with the triangular lattice (perhaps as many as $(1-o(1))n$).
Erd\H{o}s \cite{Er94b} wrote 'I could not prov... | open | L1: Tractable | 1 | Graph Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-100 | Erdős Problem #100 | Let $A$ be a set of $n$ points in $\mathbb{R}^2$ such that all pairwise distances are at least $1$ and if two distinct distances differ then they differ by at least $1$. Is the diameter of $A$ $\gg n$? | Perhaps the diameter is even $\geq n-1$ for sufficiently large $n$. Piepmeyer has an example of $9$ such points with diameter $<5$. Kanold proved the diameter is $\geq n^{3/4}$. The bounds on the distinct distance problem [89] proved by Guth and Katz \cite{GuKa15} imply a lower bound of $\gg n/\log n$.
References
[Gu... | open | L1: Tractable | 1 | Geometry | Erdős Problems | null | null | 0 | 0 | train |
EP-101 | Erdős Problem #101 | Given $n$ points in $\mathbb{R}^2$, no five of which are on a line, the number of lines containing four points is $o(n^2)$. | There are examples of sets of $n$ points with $\sim n^2/6$ many collinear triples and no four points on a line. Such constructions are given by Burr, Gr"{u}nbaum, and Sloane \cite{BGS74} and F"{u}redi and Pal\'{a}sti \cite{FuPa84}.
Gr"{u}nbaum \cite{Gr76} constructed an example with $\gg n^{3/2}$ such lines. Erd\H{o}s ... | open | L1: Tractable | 1 | Combinatorics | Erdős Problems | null | null | 0 | 0 | train |
EP-102 | Erdős Problem #102 | Let $c>0$ and $h_c(n)$ be such that for any $n$ points in $\mathbb{R}^2$ such that there are $\geq cn^2$ lines each containing more than three points, there must be some line containing $h_c(n)$ many points. Estimate $h_c(n)$. Is it true that, for fixed $c>0$, we have $h_c(n)\to \infty$? | A problem of Erd\H{o}s and Purdy. It is not even known if $h_c(n)\geq 5$ (see [101]).
It is easy to see that $h_c(n) \ll_c n^{1/2}$, and Erd\H{o}s at one point \cite{Er95} suggested that perhaps a similar lower bound $h_c(n)\gg_c n^{1/2}$ holds. Zach Hunter has pointed out that this is false, even replacing $>3$ points... | open | L1: Tractable | 1 | Combinatorics | Erdős Problems | null | null | 0 | 0 | train |
EP-103 | Erdős Problem #103 | Let $h(n)$ count the number of incongruent sets of $n$ points in $\mathbb{R}^2$ which minimise the diameter subject to the constraint that $d(x,y)\geq 1$ for all points $x
eq y$. Is it true that $h(n)\to \infty$? | It is not even known whether $h(n)\geq 2$ for all large $n$.
See also [99].",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** The number of incongruent diameter-minimizing unit-... | open | L1: Tractable | 1 | Geometry | Erdős Problems | null | null | 0 | 0 | train |
EP-104 | Erdős Problem #104 | Given $n$ points in $\mathbb{R}^2$ the number of distinct unit circles containing at least three points is $o(n^2)$. | In \cite{Er81d} Erd\H{o}s proved that $\gg n$ many circles is possible, and that there cannot be more than $O(n^2)$ many circles. The argument is very simple: every pair of points determines at most $2$ unit circles, and the claimed bound follows from double counting. Erd\H{o}s claims in a number of places this produce... | open | L1: Tractable | 1 | Graph Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-108 | Erdős Problem #108 | For every $r\geq 4$ and $k\geq 2$ is there some finite $f(k,r)$ such that every graph of chromatic number $\geq f(k,r)$ contains a subgraph of girth $\geq r$ and chromatic number $\geq k$? | Conjectured by Erd\H{o}s and Hajnal. R"{o}dl \cite{Ro77} has proved the $r=4$ case (see [923]). The infinite version (whether every graph of infinite chromatic number contains a subgraph of infinite chromatic number whose girth is $>k$) is also open.
In \cite{Er79b} Erd\H{o}s also asks whether $ \lim_{k\to \infty}\frac... | open | L1: Tractable | 1 | Graph Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-111 | Erdős Problem #111 | If $G$ is a graph let $h_G(n)$ be defined such that any subgraph of $G$ on $n$ vertices can be made bipartite after deleting at most $h_G(n)$ edges.
What is the behaviour of $h_G(n)$? Is it true that $h_G(n)/n\to \infty$ for every graph $G$ with chromatic number $\aleph_1$? | A problem of Erd\H{o}s, Hajnal, and Szemer\'{e}di \cite{EHS82}. Every $G$ with chromatic number $\aleph_1$ must have $h_G(n)\gg n$ since $G$ must contain, for some $r$, $\aleph_1$ many vertex disjoint odd cycles of length $2r+1$.
On the other hand, Erd\H{o}s, Hajnal, and Szemer\'{e}di proved that there is a $G$ with ch... | open | L1: Tractable | 1 | Graph Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-112 | Erdős Problem #112 | Let $k=k(n,m)$ be minimal such that any directed graph on $k$ vertices must contain either an independent set of size $n$ or a transitive tournament of size $m$. Determine $k(n,m)$. | A problem of Erd\H{o}s and Rado \cite{ErRa67}, who showed $k(n,m) \ll_m n^{m-1}$, or more precisely, $ k(n,m) \leq \frac{2^{m-1}(n-1)^m+n-2}{2n-3}. $ Larson and Mitchell \cite{LaMi97} improved the dependence on $m$, establishing in particular that $k(n,3)\leq n^{2}$. Zach Hunter has observed that $ R(n,m) \leq k(n,m)\l... | open | L1: Tractable | 1 | Number Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-114 | Erdős Problem #114 | If $p(z)\in\mathbb{C}[z]$ is a monic polynomial of degree $n$ then is the length of the curve $\{ z\in \mathbb{C} : \lvert p(z)\rvert=1\}$ maximised when $p(z)=z^n-1$? | A problem of Erd\H{o}s, Herzog, and Piranian \cite{EHP58}. It is also listed as Problem 4.10 in \cite{Ha74}, where it is attributed to Erd\H{o}s.
Let the maximal length of such a curve be denoted by $f(n)$.
{UL}
{LI}The length of the curve when $p(z)=z^n-1$ is $2n+O(1)$, and hence the conjecture implies in particular t... | open | L1: Tractable | 1 | Graph Theory | Erdős Problems | null | null | 0 | 0 | train |
EP-117 | Erdős Problem #117 | Let $h(n)$ be minimal such that any group $G$ with the property that any subset of $>n$ elements contains some $x
eq y$ such that $xy=yx$ can be covered by at most $h(n)$ many Abelian subgroups.
Estimate $h(n)$ as well as possible. | Pyber \cite{Py87} has proved there exist constants $c_2>c_1>1$ such that $c_1^n<h(n)<c_2^n$. Erd\H{o}s \cite{Er97f} writes that the lower bound was already known to Isaacs.
References
[Er97f] Erd\H{o}s, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10.
[Py87] Pyber... | open | L1: Tractable | 1 | Combinatorics | Erdős Problems | null | null | 0 | 0 | train |
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