Conjectures about Latin Squares
Conjecture 3.2 in [Wa2011]: Each Latin square of odd order has at least one transversal.
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Formal Conjectures: open problems with Lean statements
Formal Conjectures is an open repository, started by Google DeepMind, of conjectures stated in Lean 4 with Mathlib. Every open problem from it that we import keeps its exact Lean statement, so a proof submitted here is checked by the Lean kernel against that statement. The collection covers Erdős problems, OEIS conjectures, Ben Green's open problems, Wikipedia's lists of unsolved problems, MathOverflow questions and more.
Source: google-deepmind/formal-conjectures. Licence: Apache License 2.0.
Conjectures about Mersenne primes
For any odd natural number p if two of the following conditions hold, then all three must hold: 1. 2^p-1 is prime 2. (2^p+1)/3 is prime 3. Exists a number k such that p = 2^k pm 1 or p = 4^k pm 3
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Conjectures about the Mandelbrot and Multibrot sets
The MLC conjecture, stating that the mandelbrot set is locally connected.
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Conjectures about Weakly First Countable spaces
Problem 2 in [Ar2013]: Give an example in ZFC of a weakly first- countable compact Hausdorff space X such that 𝔠 < |X|. Note: [Ar2013] uses a blanket convention that all spaces are Tychonoff and "compact" means compact Hausdorff.
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Conjectures around homogeneous topological spaces
Problem 13 in [Ar2013]: Is it true that every infinite homogeneous compact hausdorff space contains a non-trivial convergent sequence?
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Conjectures associated with A038552
All terms of A038552 are congruent to 19 pmod24.
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Conjectures associated with A063880
All members of the sequence satisfy n ≡ 108 pmod216.
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Conjectures associated with A067720
For members of the sequence other than 8, we have k + 1 is prime.
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Conjectures associated with A100474
After a(2) = 5, is there another prime?
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Conjectures associated with A100800
A100800 Conjecture: No term is zero.
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Conjectures associated with A101779
It is conjectured k always exists.
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Conjectures associated with A105210
Cormier and Selfridge found 5 starting values for which the sequences appear to not merge. The sequences were checked up to 10^8.
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Conjectures associated with A109227
Conjecture: a(2) and a(121) are primes. Are there any more?
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Conjectures associated with A109671
Does the sequence contain every positive integer (cf. A169741)?
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Conjectures associated with A109845
Conjecture: There are infinitely many primes in this sequence.
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Conjectures associated with A109905
a(n) = 0 for n = 1, 6, 30 and 54. Are there any others?
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Conjectures associated with A109908
Conjecture: a(n) > 0 for n > 3.
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Conjectures associated with A109909
Conjecture: a(n) > 0 for n > 3.
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Conjectures associated with A110854
Do the absolute values cover A004275? A004275 is 1 together with the nonnegative even numbers. The conjecture asks whether every member of A004275 occurs as |a(n)| for some term of the sequence.
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Convergence of the Flint Hills and Cookson Hills series
The Flint Hills series summing csc(n)^2 / n^3 from n=1 to ∞ converges. (Note that we 0-index the series below.)
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Conway's 99-graph problem
Does there exist an undirected graph with 99 vertices, in which each two adjacent vertices have exactly one common neighbor, and in which each two non-adjacent vertices have exactly two common neighbors?
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Cuban Primes
This sequence is believed to be infinite.
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Cunningham chains — Jones's conjecture
Jones's conjecture (first kind): for every positive integer k, there are infinitely many primes p that start a first-kind Cunningham chain of exactly length k.
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De Giorgi's conjecture
De Giorgi's conjecture holds in dimension n ≤ 8.
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Dean's conjecture on cycles of length divisible by k
Conjecture 1.1 (Dean, 1988). For every integer k ≥ 3, every finite simple graph with minimum degree at least k contains a cycle whose length is divisible by k. A cycle has length at least 3, so the divisor is never 0 and the statement is not satisfied for a trivial reason.
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Decimal encoding of the prime factorization of n
"a(31) = a(177147) = 311. Is there any solution to a(n) = n? - _Franklin T. Adams-Watters_, Dec 18 2006"
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Dedekind Numbers
No closed-form expression that allows efficient computation of Dedekind numbers is currently known.
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Denominator of sum of reciprocals of divisors
If a(n) is in A005153, then n is in A005153. - Jaycob Coleman, Sep 27 2014 We require 0 < n because a(0) = 1 is in A005153 (practical numbers), but 0 is not.
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Denominator of Σ_k=1^n k^μ(k)
Conjecture: a(n) = primorial(n) for infinitely many n.
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Denominators of coefficients in Stirling's expansion for log(Γ(z))
Conjecture I: if n > 2, then a(A005382(n))/12 is prime, where A005382 is the sequence of primes p such that 2p-1 is also prime. Since A005382(1) = 2, A005382(2) = 3 and A005382(3) = 7, this says that a(p)/12 is prime for every prime p > 3 such that 2p-1 is also prime.
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Determinant of Hankel matrix of the first 2n-1 prime numbers
"I conjecture that a(4) is the only zero. - _Jon Perry_, Mar 22 2004" Stated as a biconditional: the claim that a(4) is the only zero asserts both that a(4) = 0 and that no other index vanishes. A bare implication a n = 0 → n = 4 would be satisfied vacuously by a sequence with no zero at all.
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Determinant of matrix with entries indicating primality of i^2 + j^2
Conjecture: a(n) = 0 for no n > 28. - _Zhi-Wei Sun_, Aug 26 2013
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Determinantal conjecture
Does the determinant of the sum A + B of two n × n normal complex matrices A and B always lie in the convex hull of the n! points Π_i (λ(A)_i + λ(B)_σ(i))? Here the numbers λ(A)_i and λ(B)_i are the eigenvalues of A and B, and σ is an element of the symmetric group S_n.
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Determinants of 2 X 2 matrices of non-overlapping blocks of 4 consecutive primes
This suggests the ratio is approaching a limit close to 0.87. Formalized as: The sequence of ratios P(N)/Neg(N) converges to a limit L, and L is in the interval (0.8, 0.9).
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Dickson's conjecture
Dickson's conjecture If a finite set of linear integer forms f_i(n) = a_i n+b_i satisfies Schinzel condition, there exist infinitely many natural numbers m such that f_i(m) are primes for all i.
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Difference between first odd semiprime > 2^n and 2^n
In this powers of 2 sequence, does 1 occur infinitely often?
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Difference of digit sums in base 3 and base 2
"This sequence is positive on average, since 1/log(3) > 1/log(4). Do all integers appear infinitely often?" - Charles R Greathouse IV, Feb 07 2013
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Digit 2 in base 3 representation of 2^n
For n > 8, 2^n is not the the sum of distinct powers of 3. Expressed here in terms of the base 3 digits of n. This conjecture is equivalent to the halting of a 15-state 2-symbol Turing Machine. TODO(lezeau): Formalize the Turing Machine version of this problem.
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Digital sum of the Fermat number 2^2^n + 1
a(0), a(1), a(5), a(6), a(7) and a(11) are primes. Are there any more?
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Diophantine m-tuples
The "strong Diophantine 5-tuple conjecture", so-called because it implies the Diophantine 5-tuple theorem (see noIntegralDiophantineFiveTuple_of_hasUniqueExtensionOfForall). [Du]
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Divisibility of 2^n + 1 by n
All members of the sequence A56777 come from prime quadruples.
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Dubner's conjecture
Every even number greater than 4208 is the sum of two twin primes.
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Elliott–Halberstam conjecture
The Elliott–Halberstam conjecture: for every θ < 1 and A > 0 there exists a constant C > 0 such that Σ_1 ≤ q ≤ x^θ E(x; q) ≤ C x/log^A x for all x > 2.
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Equational Theories
Equational Theories, Problem 8.1. Does Equation 677 imply Equation 255 in every finite magma? The project tentatively conjectures that the answer is no; a false answer is equivalent to the existence of a finite countermodel satisfying Equation 677 but not Equation 255.
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Equidistributed Sequences
The sequence (3/2)^n is equidistributed modulo 1.
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Erdős Problem #10
Is there some k such that every large integer is the sum of a prime and at most k powers of 2?
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Erdős Problem #100
Is the diameter of A at least Cn for some constant C > 0?
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Erdős Problem #1002
For any 0<α<1, let f(α,n)=1/log nΣ_1≤ k≤ n(1/2- α k). Does f(α,n) have an asymptotic distribution function? In other words, is there a non-decreasing function g such that g(-∞)=0, g(∞)=1, and lim_n→ ∞lvert α∈ (0,1): f(α,n)≤ crvert=g(c)?
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Erdős Problem #1003
Are there infinitely many solutions to φ(n) = φ(n+1), where φ is the Euler totient function?
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Erdős Problem #1004
For any fixed c > 0, if x is sufficiently large then there exists n ≤ x such that the values of φ(n+k) are all distinct for 1 ≤ k ≤ (log x)^c. This is an open problem.
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Erdős Problem #101
Given n points in ℝ^2, no five of which are on a line, the number of lines containing four points is o(n^2).
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Erdős Problem #102
Let c > 0 and let h_c(n) be such that for any n points in ℝ^2 with at least cn^2 lines that each contain more than three of the points, some line contains h_c(n) of the points. Is it true that, for fixed c > 0, h_c(n) → ∞?
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Erdős Problem #1020
Let f(n;r,k) be the maximal number of edges in an r-uniform hypergraph which contains no set of k many independent edges. For all r≥ 3, f(n;r,k)=max(C(rk-1, r), C(n, r)-C(n-k+1, r)). Note: the source states the formula with no range on n or k, but some restriction is needed: e.g.
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Erdős Problem #1029
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 R(k)/k2^k/2→ ∞.
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Erdős Problem #103
Let h(n) count the number of incongruent sets of n points in ℝ^2 which minimise the diameter subject to the constraint that d(x,y)≥ 1 for all points x≠ y. Is it true that h(n)→ ∞?
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Erdős Problem #1030
Let R(k,l) be the usual Ramsey number: the smallest n such that if the edges of K_n are coloured red and blue then there exists either a red K_k or a blue K_l. Prove the existence of some c>0 such that lim_k→ inftyR(k+1,k)/R(k,k)> 1+c. A problem of Erdős and Sós.
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Erdős Problem #1035
Is there a constant c > 0 such that every graph on 2^n vertices with minimum degree > (1-c) · 2^n contains the n-dimensional hypercube Q_n? This is Erdős's question [Er93, p. 345]. See also [576] for the extremal number of edges that guarantee a Q_n.
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Erdős Problem #1038
What is the infimum of |x ∈ ℝ : |f x| < 1| over all nonconstant monic polynomials f such that all of its roots are real and contained in [-1,1]?
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Erdős Problem #104
Given n points in ℝ^2 the number of distinct unit circles containing at least three points is o(n^2).
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Erdős Problem #1049
Let t>1 be a rational number. Is Σ_n=1^∞1/t^n-1=Σ_n=1^∞ τ(n)/t^n irrational, where τ(n) counts the divisors of n? A conjecture of Chowla.
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Erdős Problem #1052
Are there only finitely many unitary perfect numbers?
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Erdős Problem #1054
Let f(n) be the minimal integer m such that n is the sum of the k smallest divisors of m for some k≥ 1. Is it true that f(n)=o(n)?
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Erdős Problem #1055
A prime p is in class 1 if the only prime divisors of p+1 are 2 or 3. In general, a prime p is in class r if every prime factor of p+1 is in some class ≤ r-1, with equality for at least one prime factor. Are there infinitely many primes in each class?
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Erdős Problem #1056
Let k ≥ 2. Does there exist a prime p and consecutive intervals I_0,…,I_k such that Πlimits_n∈I_in ≡ 1 mod n for all 1 ≤ i ≤ k?
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Erdős Problem #1057
Is it true that C(x)=x^1-o(1)? This is discussed in problem A13 of Guy's collection [Gu04].
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Erdős Problem #1059
Are there infinitely many primes p such that p - k! is composite for each k such that 1 ≤ k! < p?
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Erdős Problem #1060
The conjecture is about the function f(n) which counts the number of solutions to kσ(k)=n, where σ(k) is the sum of divisors of k. The first bound is that f(n) grows slower than any power of n^(1/loglog n). The second bound is that f(n) is at most a power of log n.
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Erdős Problem #1061
How many (ordered) solutions are there to σ(a) + σ(b) = σ(a + b) with a + b ≤ x? Is it true that this number is asymptotic to c * x for some constant c > 0?
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Erdős Problem #1062
Erdős asked whether the limiting density f n / n exists and, if so, whether it is irrational.
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Erdős Problem #1063
Estimate n_k by finding a better upper bound than Cambie's n_k ≤ k · lcm(1, dotsc, k-1). The comparator takes its least common multiple in ℕ and casts the result.
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Erdős Problem #1065
Are there infinitely many primes p such that p = 2^k q + 1 for some prime q and k ≥ 0? This is mentioned as B46 in Unsolved Problems in Number Theory by Richard K. Guy*
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Erdős Problem #1068
Does every graph with chromatic number aleph_1 contain a countable subgraph which is infinitely connected?
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Erdős Problem #1072
Is it true that there are infinitely many p for which f(p) = p − 1?
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Erdős Problem #1073
Is it true that A(x) ≤ x^o(1)?
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Erdős Problem #1074
Let S be the set of all m≥ 1 such that there exists a prime pnot≡ 1pmodm such that m! + 1 ≡ 0pmodp. Does lim|S∩[1, x]|/x exist?
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Erdős Problem #108
For every r ≥ 4 and k ≥ 2 is there some finite f(k,r) such that every graph of chromatic number ≥ f(k,r) contains a subgraph of girth ≥ r and chromatic number ≥ k?
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Erdős Problem #1082
Let A⊂ ℝ^2 be a set of n points with no three on a line. Does A determine at least ⌊ n/2⌋ distinct distances?
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Erdős Problem #1083
Let d≥ 3, and let f_d(n) be the minimal m such that every set of n points in ℝ^d determines at least m distinct distances. Estimate f_d(n) - in particular, is it true that f_d(n)=n^2/d-o(1)?
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Erdős Problem #1088
Let f_d(n) be the minimal m such that any set of m points in ℝ^d contains a set of n points for which any two determined distances are distinct. Erdős Problem 1088 asks to estimate f_d(n). In particular, is it true that, for every fixed n ≥ 3, f_d(n) = 2^o(d) as d → ∞?
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Erdős Problem #1093
Are there infinitely many binomial coefficients with deficiency 1?
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Erdős Problem #1094
For all n≥ 2k the least prime factor of C(n, k) is ≤max(n/k,k), with only finitely many exceptions.
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Erdős Problem #11
Is every odd n > 1 the sum of a squarefree number and a power of 2?
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Erdős Problem #1101
1. There is NO good sequence with polynomial growth.
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Erdős Problem #1106
Let p(n) be the partition number of n and F(n) be the number of distinct prime factors of ∏_i= 1 ^ n p(n), then F(n) tends to infinity when n tends to infinity.
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Erdős Problem #1107
Let r ≥ 2. Is every large integer the sum of at most r + 1 many r-powerful numbers?
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Erdős Problem #1108
For each k ≥ 2, does the set A = Σ_n∈ Sn! : S⊂ ℕ finite of all finite sums of distinct factorials contain only finitely many k-th powers?
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Erdős Problem #1109
Let f(N) be the size of the largest subset A⊆ 1,…,N such that every n∈ A+A is squarefree. Estimate f(N). In particular, is it true that f(N)≤ N^o(1), or even f(N) ≤ (log N)^O(1)? This theorem formalizes the subpolynomial bound as f(N) = O(N^ε) for every ε > 0.
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Erdős Problem #1110
Let p>q≥ 2 be two coprime integers. We call n representable if it is the sum of integers of the form p^kq^l, none of which divide each other. If p,q≠ 2,3 then what can be said about the density of non-representable numbers?
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Erdős Problem #1113
Erdős Problem 1113. Do there exist Sierpiński numbers that possess no finite covering set of primes? Erdős and Graham [ErGr80] conjectured that the answer is yes. A negative answer would imply that there are infinitely many Fermat primes.
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Erdős Problem #1133
Let C>0. There exists ε>0 such that if n is sufficiently large the following holds. For any x_1,…,x_n∈ [-1,1] there exist y_1,…,y_n∈ [-1,1] such that, if P is a polynomial of degree m<(1+ε)n with P(x_i)=y_i for at least (1-ε)n many 1≤ i≤ n, then max_x∈ [-1,1]lvert P(x)rvert >C.
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Erdős Problem #1135
The Collatz conjecture states that for any positive integer n, there exists a natural number m such that the m-th term of the sequence is 1.
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Erdős Problem #1137
Let d_n=p_n+1-p_n, where p_n denotes the nth prime. Is it true that max_n < xd_nd_n-1/(max_n < xd_n)^2→ 0 as x→ ∞?
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Erdős Problem #1139
Let 1≤ u_1 < u_2 < ⋯ be the sequence of integers with at most 2 prime factors. Is it true that limsup_k → ∞ u_k+1-u_k/log k=∞?
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Erdős Problem #1142
Are there infinitely many n > 2 such that n - 2^k is prime for all k ≥ 1 with 2^k < n? The only known such n are 4, 7, 15, 21, 45, 75, 105 (OEIS A039669).
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Erdős Problem #1145
Let A=1≤ a_1 < a_2 < ⋯ and B=1≤ b_1 < b_2 < ⋯ be sets of integers with a_n/b_n→ 1. If A+B contains all sufficiently large positive integers then is it true that limsup 1_Aast 1_B(n)=∞? A conjecture of Erdős and Sárközy.
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Erdős Problem #1146
Is B=2^m3^n : m,n≥ 0 an essential component? In [Ru99] Ruzsa states "The simplest set with a chance to be an essential component is the collection of numbers in the form 2^m3^n and Erdős often asked whether it is an essential component or not; I do not even have a plausible guess."
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Erdős Problem #1150
Is there some constant c > 0 such that, for all large enough n and all polynomials P of degree n with coefficients in -1, 1, max_|z|=1 |P(z)| > (1 + c) √(n)?
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Erdős Problem #1159
Determine whether there exists a constant C>1 such that the following holds. Let P be a finite projective plane. Must there exist a set of points S such that 1≤ lvert S∩ ℓrvert ≤ C for all lines ℓ?
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Erdős Problem #1167
Erdős Problem 1167. Let r ≥ 2 be finite, γ ≥ 2, and λ be an infinite cardinal. Let κ_α > r be cardinals for all α < γ. Is it true that 2^λ → (κ_α + 1)_α < γ^r+1 implies λ → (κ_α)_α < γ^r? Here + means cardinal addition, so that κ_α + 1 = κ_α if κ_α is infinite. A problem of Erdős, Hajnal, and Rado.
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Erdős Problem #1175
Let κ be an uncountable cardinal. Must there exist a cardinal λ such that every graph with chromatic number λ contains a triangle-free subgraph with chromatic number κ? Shelah proved that a negative answer is consistent when κ = λ = aleph_1 (see erdos_1175.variants.aleph_one).
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