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1011 problems

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
Source Any Curated here 10Formal Conjectures (Lean) 464Optimization constants 24AlphaEvolve problems 2
Verification AnyA · Machine-checkableB · ReproducibleC · Reviewed Tier AnyStandardHardGrand challenges

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A Hard Number theory · Formal Conjectures (Lean)

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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A Hard Number theory · Formal Conjectures (Lean)

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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A Hard Number theory · Formal Conjectures (Lean)

Cuban Primes

This sequence is believed to be infinite.

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A Hard Number theory · Formal Conjectures (Lean)

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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A Hard Number theory · Formal Conjectures (Lean)

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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A Hard Number theory · Formal Conjectures (Lean)

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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A Hard Number theory · Formal Conjectures (Lean)

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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A Hard Number theory · Formal Conjectures (Lean)

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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A Hard Number theory · Formal Conjectures (Lean)

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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A Hard Number theory · Formal Conjectures (Lean)

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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A Hard Number theory · Formal Conjectures (Lean)

Divisibility of 2^n + 1 by n

All members of the sequence A56777 come from prime quadruples.

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A Hard Number theory · Formal Conjectures (Lean)

Dubner's conjecture

Every even number greater than 4208 is the sum of two twin primes.

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A Hard Number theory · Formal Conjectures (Lean)

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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A Hard Number theory · Formal Conjectures (Lean)

Equidistributed Sequences

The sequence (3/2)^n is equidistributed modulo 1.

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A Hard Number theory · Formal Conjectures (Lean)

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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A Hard Number theory · Formal Conjectures (Lean)

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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A Hard Number theory · Formal Conjectures (Lean)

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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A Hard Number theory · Formal Conjectures (Lean)

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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A Hard Number theory · Formal Conjectures (Lean)

Erdős Problem #1052

Are there only finitely many unitary perfect numbers?

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A Hard Number theory · Formal Conjectures (Lean)

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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A Hard Number theory · Formal Conjectures (Lean)

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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A Hard Number theory · Formal Conjectures (Lean)

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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A Hard Number theory · Formal Conjectures (Lean)

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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A Hard Number theory · Formal Conjectures (Lean)

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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A Hard Number theory · Formal Conjectures (Lean)

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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A Hard Number theory · Formal Conjectures (Lean)

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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A Hard Number theory · Formal Conjectures (Lean)

Erdős Problem #1062

Erdős asked whether the limiting density f n / n exists and, if so, whether it is irrational.

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A Hard Number theory · Formal Conjectures (Lean)

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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A Hard Number theory · Formal Conjectures (Lean)

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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A Hard Number theory · Formal Conjectures (Lean)

Erdős Problem #1072

Is it true that there are infinitely many p for which f(p) = p − 1?

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A Hard Number theory · Formal Conjectures (Lean)

Erdős Problem #1073

Is it true that A(x) ≤ x^o(1)?

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A Hard Number theory · Formal Conjectures (Lean)

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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