abc conjecture
For every positive real number ε, there exist only finitely many triples (a, b, c) of coprime positive integers, with a + b = c, such that c > rad(abc)^(1+ε)
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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.
All fields Mathematics 936Computer science 28Physics & astronomy 22Chemistry & materials 13Life sciences 6Earth & society 6
Combinatorics 158Number theory 500Graph theory 71Algebra 45Geometry 72Analysis 69Probability 11Logic & formalisation 10
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Agoh-Giuga conjecture
The Agoh-Giuga Conjecture, Agoh's formulation
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Agrawal's conjecture
Agrawal's Primality Conjecture. Does the congruence (X-1)^n ≡ X^n - 1 pmodn, X^r-1 imply n is prime (with a specific exception for n^2 ≡ 1 pmodr)? While the "if" direction is a known theorem, the "only if" direction remains a conjecture.
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Alternating sum of signs of powers of 3/2
For n large enough, does a(n) > √(n) always hold?
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Amicable numbers
Relatively prime amicable numbers conjecture. Do there exist amicable numbers (a, b) with gcd(a, b) = 1? All known amicable pairs share a common factor. It is an open question whether a pair of relatively prime amicable numbers can exist. Reference: Wikipedia
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An Arithmetic Sum Associated with the Classical Theta Function
Conjecture 1.1: For any odd prime k, the sum associated with the classical theta function θ_3, S(k) is positive.
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Andrica's conjecture
Andrica's conjecture The inequality √(p_n+1)-√(p_n) < 1 holds for all n, where p_n is the n-th prime number.
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Apéry numbers
For each n = 1, 2, 3, … the polynomial a_n(x) = Σ_k=0^n C(n, k)^2 C(n+k, k) x^k is irreducible over the field of rational numbers. - Zhi-Wei Sun, Mar 21 2013
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Are prime numbers among sums of prime numbers distributed as frac n2ln(n)?
The conjecture claims that π_n∼frac n2ln(n). In other words, primes are distributed among the much sparser sequence (S_n)_n with essentially the same density as in the positive integers, up to a factor of 2. MathOverflow 434111.
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Array read by upward antidiagonals
A "Goldbach Conjecture" for this sequence: when there are n terms between consecutive odd integers 2n+1 and 2n+3 for n > 0, at least one will be the product of 2 primes (not necessarily distinct).
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Artin's conjecture on primitive roots
Artin's Conjecture on Primitive Roots, first half. Let a be an integer that is not a square number and not −1. Then the set S(a) of primes p such that a is a primitive root modulo p has a positive asymptotic density inside the set of primes. In particular, S(a) is infinite.
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Ascending descending base exponent transform of 2^n
The first prime terms in this (always odd) sequence are a(1) = 3, a(3) = 41, and a(4) = 593. What is the next prime? The OEIS comment currently says a(5) = 543, but this conflicts with its defining formula, b-file, and examples: the actual index-five term is the composite number 135457.
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Ascending descending base exponent transform of factorials
Is there a nontrivial power after a(4) = 5^3?
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Ascending descending base exponent transform of squares
The smallest prime in this sequence is a(2) = 5. What is the next prime?
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Asymptotic density of powerful numbers
Can the exponent 1/6 in the error term of the Bateman–Grosswald asymptotic be improved unconditionally? That is, is there δ > 0 such that Q(x) = ζ(3/2)/ζ(3) x^1/2 + ζ(2/3)/ζ(2) x^1/3 + O(x^1/6 - δ)? Improvements are known under the Riemann Hypothesis.
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Balanced prime conjecture
Let p_k be the k-th prime number. Are there infinitely many n such that (p_n + p_n+2) / 2 is prime?
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Bateman-Horn Conjecture
The Bateman-Horn Conjecture Given a finite collection of distinct irreducible polynomials non-constant f_1, f_2, …, f_k ∈ ℤ[x] with positive leading coefficients that satisfy the Schinzel condition, the number of positive integers n ≤ x for which all polynomials f_i are simultaneously prime is…
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Beal conjecture
The Beal Conjecture: if we are given positive integers A, B, C, x, y, z such that x, y, z > 2 and A^x + B^y = C^z then A, B, C have a common divisor.
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Ben Green's Open Problem 2
Let A ⊂ ℤ be a set of n integers. Is there a set S ⊂ A of size (log n)^100 such that the restricted sumsetS hat+ S is disjoint from A?
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Ben Green's Open Problem 45
Can we pick residue classes a_p pmodp, one for each prime p ≤ N, such that every integer ≤ N lies in at least 10 of them? Erdős remarks that he does not know how to answer it with 10 replaced by 2; this is Erdos689.erdos_689.
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Ben Green's Open Problem 46
We conjecture that the best-known lower bound can be improved.
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Ben Green's Open Problem 60
Is there an absolute constant c > 0 such that, whenever A ⊆ ℕ is a set of squares with |A| ≥ 2, the sumset A + A satisfies |A + A| ≥ |A|^1 + c?
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Ben Green's Open Problem 61
Suppose that A + A contains the first n squares. Is |A| ≥ n^1 - o(1)? It is known that necessarily |A| ≥ n^2/3 - o(1), whilst in the other direction there do exist such A with |A| ≪_C n / log^C n for any C.
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Ben Green's Open Problem 62
Let p be a large prime, and let A be the set of all primes less than p. Is every x ∈ 1, …, p-1 congruent to some product a_1 a_2 where a_1, a_2 ∈ A?
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Ben Green's Open Problem 66
Is there always a sum of two squares between X - 1/10X^1/4 and X? We formalize this as an eventual statement for sufficiently large real X.
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Ben Green's Open Problem 82
Let A ⊂ ℤ be a set of size n. For how many θ ∈ ℝ/ℤ must we have Σ_a ∈ A cos(2π aθ) = 0? The answer is the function minZeros.
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Betrothed numbers
Same parity betrothed numbers conjecture. Do there exist betrothed numbers (m, n) where both have the same parity (both even or both odd)? All known betrothed pairs consist of one even and one odd number. The requirement m ≠ n is part of the question: IsBetrothed n n says σ(n) = 2n + 1, i.e.
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Binary representation of primes that divide a number, in decimal
Starting at any n and iterating the map n ↦ a(n), we will always reach 0. - _Antti Karttunen_, Jun 18,20 2017
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Brocard's Conjecture
Brocard's Conjecture For every n ≥ 2, between the squares of the n-th and (n+1)-th primes, there are at least four prime numbers.
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Büchi's problem
Büchi's problem There exists a positive integer M such that, for all integers x and a, if (x+n)^2 + a is a square for M consecutive values of n, then a = 0.
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Bugeaud Collection of Conjectures and Open Questions: Confined Powers of Non-Pisot Numbers
Problem 10.7. Let ε be a positive real number. Are there arbitrarily large real numbers α such that α is not a Pisot number and all the fractional parts α^n, n ≥ 1, are lying in an interval of length ε / α? [Bug12b]
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Bugeaud Collection of Conjectures and Open Questions: Fractional Parts of Powers
Problem 10.1. Are there a transcendental number α and a positive real number ξ such that lVert ξ α^n rVert tends to~0 as~n tends to infinity? [Har19] (Trivial for |α| < 1)
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Bugeaud Collection of Conjectures and Open Questions: Mahler's Z-numbers
Problem 10.9. There are no real numbers ξ such that 0 ≤ ξ (3/2)^n < 1/2 for every positive integer n, i.e. no Z-number exists. Posed by Mahler [Mah68].
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Bugeaud Collection of Conjectures and Open Questions: p-adic Littlewood Conjecture
Problem 10.8 (p-adic Littlewood conjecture). For every real number ξ and every prime number p, inf_q ≥ 1 q · lVert q ξ rVert · |q|_p = 0, where lVert · rVert denotes the distance to the nearest integer and |·|_p denotes the p-adic absolute value. Posed by de Mathan and Teulié [dMT04].
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Bugeaud Collection of Conjectures and Open Questions: Pisot orbits on the Cantor set
Problem 10.61. Let α > 2 be a Pisot number. For every ξ ∈ C(α) the sequence (ξ α^n)_n ≥ 1 is not uniformly distributed modulo one.
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Bunyakovsky conjecture
Bunyakovsky conjecture If a polynomial f over integers satisfies both Schinzel and Bunyakovsky conditions, there exist infinitely many natural numbers m such that f(m) is prime.
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Can a prime p satisfy 2^p-1 ≡ 1 pmodp^2 and 3^p-1 ≡ 1 pmodp^2?
Can a prime p satisfy 2^p-1 ≡ 1 pmodp^2 and 3^p-1 ≡ 1 pmodp^2 simultaneously? That is, does there exist a prime p that is both a Wieferich prime and a Mirimanoff prime? Wikipedia's list of unsolved problems poses this question, citing J. B. Dobson, On Lerch's formula for the Fermat quotient.
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Carmichael's totient function conjecture
Carmichael's totient function conjecture: For every positive natural number n, there exists a natural number m with m ≠ n, such that φ(n) = φ(m).
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Catalan-Mersenne numbers
Catalan-Mersenne conjecture: All terms of the Catalan-Mersenne sequence are prime.
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Catalan's conjecture and related Diophantine equations
For positive integers a, b, and c, there are only finitely many positive solutions (x, y, m, n) to the equation ax^n - by^m = c where (m, n) ≠ (2, 2) and x, y > 1.
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Central binomial sum a(n) = Σ_k=0^n (-4)^k C(n, k)^2 C(n-k, k)^2
If p is a prime with p ≡ 1, 9 pmod20 and p = x^2 + 5y^2 with x, y integers, then Σ_k=0^p-1 a(k) ≡ 4x^2 - 2p pmodp^2. - _Zhi-Wei Sun_, Jul 01 2010
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Central binomial sum a(n) = Σ_k=0^n C(n, k)^2 C(n-k, k)^2 (-16)^k
If p is a prime with (p/7) = 1 and p = x^2 + 7y^2 with x, y integers, then Σ_k=0^p-1 (-1)^k a(k) ≡ 4x^2 - 2p pmodp^2. - _Zhi-Wei Sun_, Jul 17 2010
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Central trinomial coefficients
An integer n > 3 is prime if and only if a(n) ≡ 1 pmodn^2. We have verified this for n up to 8 · 10^5, and proved that a(p) ≡ 1 pmodp^2 for any prime p > 3 (cf. A277640). - Zhi-Wei Sun, Nov 30 2016
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Chua's Euclidean prime sequence
Does Chua's sequence contain every prime?
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Class number problem for real quadratic fields
There are infinitely many real quadratic fields ℚ(√d) with class number one, where d > 1 is a squarefree integer.
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Coefficients of Π_k>0 (1 - x^k/k!)
The coefficients c(n) of A(x)^2 = (Σ_n ≥ 0 a(n) x^n)^2 differ in sign from c(n-1) if and only if n is a triangular number. - _Peter Bala_, Mar 17 2022
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Collatz step differences
Conjecture 1: More than half of the terms are 0. - _Ya-Ping Lu_, May 04 2024
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Concatenation of the next n numbers
"The second term is a prime. When is the next prime, if there is another? - _N. J. A. Sloane_, Dec 16 2016"
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Congruent Number
Tunnell's theorem (sufficient condition assuming BSD) for odd squarefree congruent numbers.
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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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