Erdős Problem #826
Are there infinitely many n such that, for all k≥ 1 τ(n + k) ≪ k?
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Erdős problems (Erdos problems) you can work on with AI
Paul Erdős posed hundreds of problems, many with cash prizes. Thomas Bloom collects them at erdosproblems.com, and a community database (teorth/erdosproblems) tracks their status. Most of the open ones listed here have Lean statements in Formal Conjectures; each imported page shows the status from erdosproblems.com, the prize if there is one, and what kind of progress would count.
Source: erdosproblems.com (Thomas Bloom). Licence: Lean statements from Formal Conjectures (Apache 2.0); status data from the community database teorth/erdosproblems (Apache 2.0).
Erdős Problem #828
Is it true that, for any a ∈ ℤ, there are infinitely many n such that φ(n) | n + a?
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Erdős Problem #829
Erdős Problem 829 (open). Let A ⊆ ℕ be the set of perfect cubes. Is it true that (1_A ast 1_A)(n) ≪ (log n)^O(1)? That is, does there exist a natural number C such that the number of representations of n as a sum of two cubes is O((log n)^C) as n → ∞?
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Erdős Problem #830
Erdos Problem 830, Part 1 We say that a,b∈ ℕ are an amicable pair if σ(a)=σ(b)=a+b. Are there infinitely many amicable pairs?
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Erdős Problem #835
Does there exist a k>2 such that the k-sized subsets of 1,...,2k can be coloured with k+1 colours such that for every A⊂ 1,…,2k with lvert Arvert=k+1 all k+1 colours appear among the k-sized subsets of A?
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Erdős Problem #839
Erdős Problem 839 (Part 1) [Er78f][Er92c]: Let 1 ≤ a_1 < a_2 < ⋯ be a strictly increasing sequence of positive integers such that no a_i is the sum of consecutive a_j for j < i. Is it true that limsup a_n / n = ∞?
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Erdős Problem #849
Is it true that, for every integer t≥1, there is some integer a such that n choose k = a with 1≤ k ≤ n/2 has exactly t solutions?
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Erdős Problem #85
Is it true that, for all large n, f(n + 1) ≥ f(n)?
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Erdős Problem #850
Can there exist two distinct integers x and y such that x,y have the same prime factors, x+1,y+1 have the same prime factors, and x+2,y+2 also have the same prime factors?
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Erdős Problem #853
Let d_n = p_n+1 - p_n, where p_n is the nth prime. Let r(x) be the smallest even integer t such that d_n = t has no solutions for n ≤ x. Is it true that r(x) → ∞?
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Erdős Problem #855
Erdős Problem 855 (Segal's conjecture): π(x + y) ≤ π(x) + π(y) for all sufficiently large x, y, i.e. for all x, y ≥ N for some N.
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Erdős Problem #857
Estimate m(n,k), or better give an asymptotic formula.
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Erdős Problem #859
The density of the divisor sum set is asymptotically equivalent to c_1 / log(t)^c_2.
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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 ≥ (1/2+o(1))n2^n-1 many edges contains a C_4?
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Erdős Problem #87
Let 0 < ε < 1. Is it true that, if k is sufficiently large, then R(G) > (1-ε)^k R(k) for every graph G with chromatic number χ(G)=k? The restriction ε < 1 excludes negative bases in (1-ε)^k. This problem is #12 in Ramsey Theory in the graphs problem collection.
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Erdős Problem #872
Erdős Problem 872, part (i) (weak form): there exists a constant ε > 0 such that the game length is at least ε · n for all sufficiently large n.
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Erdős Problem #873
Let A = a_1 < a_2 < … ⊆ ℕ and let F(A,X,k) count the number of i such that [a_i,a_i+1, … ,a_i+k−1] < X, where the left-hand side is the least common multiple. Is it true that, for every ε > 0, there exists some k such that F(A,X,k) < X^ε?
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Erdős Problem #881
Let A ⊂ ℕ be an additive basis of order k which is minimal in the sense that if B ⊂ A is any infinite set, then A B is not a basis of order k. Must there exist an infinite B ⊂ A such that A B is an additive basis of order k + 1?
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Erdős Problem #883
For A⊆ 1,…,n let G(A) be the graph with vertex set A, where two integers are joined by an edge if they are coprime. Is it true that if |A| > ⌊ n/2 ⌋ + ⌊ n/3 ⌋ - ⌊ n/6 ⌋ then G(A) contains all odd cycles of length ≤ n/3 + 1? A problem of Erdős and Sárközy [ErSa97].
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Erdős Problem #885
Is it true that, for every k ≥ 1, there exist integers N_1 < … < N_k such that |∩_i D(N_i)| ≥ k?
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Erdős Problem #886
Let ε>0. Is it true that, for all large n, the number of divisors of n in (n^1/2,n^1/2+n^1/2-ε) is O_ε(1)? Erdős attributes this conjecture to Ruzsa.
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Erdős Problem #887
Is there an absolute constant K such that, for every C > 0, if n is sufficiently large then n has at most K divisors in (n^1/2, n^1/2 + C n^1/4).
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Erdős Problem #889
Let v(n,k) count the prime factors of n+k which do not divide n+i for 0≤ i < k. Is it true that v_0(n)=max_k≥ 0v(n,k)→ ∞ as n→ ∞?
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Erdős Problem #89
Erdős [Er46] asked whether every set of n distinct points in ℝ^2 determines ≫ n/√(log n) many distinct distances.
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Erdős Problem #890
If ω_k(n) counts the number of distinct prime factors of n which are >k, then is it true that, for every k≥ 1, liminf_n→ ∞Σ_0≤ i < kω_k(n+i)≤ k?
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Erdős Problem #891
Let 2=p_1 < p_2 < ⋯ be the primes and k≥ 2. Is it true that, for all sufficiently large n, there must exist an integer in [n,n+p_1⋯ p_k) with >k many prime factors?
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Erdős Problem #893
Does the limit lim_n→∞ f(2n)/f(n) tend to infinity? (Other finite limits have been ruled out by [KoLu25], see below)
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Erdős Problem #9
Is the upper density of the set of odd numbers that cannot be expressed as a prime plus two powers of 2 positive?
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Erdős Problem #906
Does there exists an entire non-zero transcendental function f : ℂ → ℂ such that for any sequence n₀ < n₁ < ..., z | ∃ k, iteratedDeriv (n k) f z = 0 is dense.
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Erdős Problem #91
Suppose A⊂ ℝ^2 has lvert Arvert=n and minimises the number of distinct distances between points in A. Prove that for large n there are at least two (and probably many) such A which are non-similar.
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Erdős Problem #912
Prove that there exists some c>0 such that h(n) ∼ c (n/log n)^1/2 as n→ ∞.
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Erdős Problem #913
Are there infinitely many n such that if n(n + 1) = Π_i p_i^k_i is the factorisation into distinct primes then all exponents k_i are distinct?
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Erdős Problem #918
Erdős Problem #918
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Erdős Problem #930
Is it true that, for every r, there is a k such that if I_1,…,I_r are disjoint intervals of consecutive integers, all of length at least k, then Π_1≤ i≤ rΠ_m∈ I_im is not a perfect power?
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Erdős Problem #931
Let k_1 ≥ k_2 ≥ 3. Are there only finitely many n_2≥ n_1 + k_1 such that Π_1≤ i≤ k_1(n_1 + i) and Π_1≤ j≤ k_2 (n_2 + j) have the same prime factors?
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Erdős Problem #932
Let p_k denote the kth prime. For infinitely many r there are at least two integers p_r < n < p_r+1 all of whose prime factors are < p_r + 1 - p_r.
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Erdős Problem #933
If n(n+1)=2^k3^lm, where (m,6)=1, then is it true that limsup_n→ ∞ 2^k3^l/nlog n=∞?
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Erdős Problem #938
Let A=n_1 < n_2 < ⋯ be the sequence of powerful numbers (if p| n then p^2| n). Are there only finitely many three-term progressions of consecutive terms n_k,n_k+1,n_k+2?
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Erdős Problem #939
If r≥4 then can the sum of r-2 coprime r-powerful numbers ever be itself r-powerful?
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Erdős Problem #940
Let r ≥ 3. Is it true that the set of integers which are the sum of at most r r-powerful numbers has density 0?
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Erdős Problem #942
Is there some constant c > 0 such that h(n) < (log n)^c + o(1) and, for infinitely many n, h(n) > (log n)^c - o(1).
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Erdős Problem #943
Let A be the set of powerful numbers. Is is true that 1_Aast 1_A(n)=n^o(1) for every n?
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Erdős Problem #944
Let k ≥ 4 and r≥ 1. Must there exist a graph G with chromatic number k such that every vertex is critical, yet every critical set of edges has size >r?
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Erdős Problem #945
Is it true that F(x) ≤ (log x)^O(1)?
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Erdős Problem #949
Let S ⊆ ℝ be a set containing no solutions to a + b = c. Must there be a set A ⊆ ℝ ∖ S of cardinality continuum such that A + A ⊆ ℝ∖ S?
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Erdős Problem #950
Is it true that liminf f(n)=1?
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Erdős Problem #951
If 1 < a 0 < ... has property Erdos951Prop, is it true that #a i ≤ x ≤ π x?
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Erdős Problem #952
Is there an infinite sequence of distinct Gaussian primes x_1,x_2,… such that lvert x_n+1-x_nrvert ≪ 1?
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Erdős Problem #955
If A⊂ ℕ has density 0 then s^-1(A) must also have density 0. A conjecture of Erdős, Granville, Pomerance, and Spiro [EGPS90].
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Erdős Problem #959
Let A⊆ ℝ^2 be a set of size n and let d_1,…,d_k be the set of distinct distances determined by A. Let f(d) be the number of times the distance d is determined, ordered so that f(d_1)≥ f(d_2)≥ ⋯ ≥ f(d_k).
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Erdős Problem #96
If n points in ℝ^2 form a convex polygon then there are O(n) many pairs which are distance 1 apart.
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Erdős Problem #961
It is conjectured that f(k) ≪ (log k)^O(1).
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Erdős Problem #962
Main conjecture: log k(n) ≤ (log n)^(1/2 + o(1))
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Erdős Problem #968
Does the set n | u n < u (n+1) have positive lower density?
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Erdős Problem #97
Does every convex polygon have a vertex with no other 4 vertices equidistant from it?
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Erdős Problem #970
Let h(k) be Jacobsthal's function, defined to as the minimal m such that, if n has at most k prime factors, then in any set of m consecutive integers there exists an integer coprime to n. Determine the order of magnitude of h(k). In particular, is it true that h(k) ≪ k^2?
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Erdős Problem #971
Let p(a, d) be the least prime congruent to a (mod d). Does there exist a constant c > 0 such that for all large d, p(a, d) > (1 + c) φ(d) log d for ≫ φ(d) many values of a?
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Erdős Problem #972
Erdős problem 972. Let α > 1 be irrational. Are there infinitely many primes p such that ⌊ pα ⌋ is also prime?
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Erdős Problem #975
For an irreducible polynomial f ∈ ℤ[x] with f(n) ≥ 1 for sufficiently large n, does there exists a constant c = c(f) > 0 such that Σ_n ≤ x τ(f(n)) ≈ c · x log x? Note that it is unclear whether the polynomial should have integer coefficients or merely be integer-valued. We assume the former.
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Erdős Problem #978
If k>3 (and k ≠ 2^l), and for all primes p there exists n such that p^k-2nmid f(n), then are there infinitely many n for which f(n) is (k-2)-power-free?
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Erdős Problem #979
Let k ≥ 2, and let f_k(n) count the number of solutions to n = p_1^k + … + p_k^k, where the p_i are prime numbers. Is it true that limsup f_k(n) = ∞?
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Erdős Problem #98
Let h(n) be such that any n points in 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→ ∞?
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Erdős Problem #982
If n distinct points in ℝ^2 form a convex polygon then some vertex has at least lfloorn/2⌋ different distances to other vertices.
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Erdős Problem #985
Is it true that, for every prime p, there is a prime q ≤ p which is a primitive root modulo p?
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Erdős Problem #99
For sufficiently large n, is it the case that any set of n points with minimum distance 1 that minimizes diameter must contain an equilateral triangle of side length 1?
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Erdős Problem #995
Erdős Problem 995: For every lacunary sequence (n_k) of integers and every f ∈ L^2([0,1]) with ∫_0^1 f = 0, is it true that for almost all α, Σ_k < N f(α n_k) = o (N √(loglog N))?
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Erdős Problem #996
Does there exists a positive constant C such that for all f ∈ L²[0,1] and all lacunary sequences n, if ‖f - fₖ‖₂ = O(1 / log log log k ^ C), then for almost every x, lim ∑ k ∈ Finset.range N, f (n k • x)) / N = ∫ t, f t ∂t?
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