Conjectures associated with A109905
a(n) = 0 for n = 1, 6, 30 and 54. Are there any others?
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Open problems
Each problem states how progress is verified and what counts as a contribution. Besides the problems curated here, the catalogue includes open conjectures from Formal Conjectures (with Lean statements), optimization constants and the AlphaEvolve problems. Know one that belongs here? Propose a problem.
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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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