Erdős Problem #138
In [Er80] Erdős asks whether lim_k → ∞ (W(k))^1/k = ∞
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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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Erdős Problem #14
Let A ⊆ ℕ. Let B ⊆ ℕ be the set of integers which are representable in exactly one way as the sum of two elements from A. Is it true that for all ε > 0 and large N, |1,…,N ∖ B| ≫_ε N^1/2 - ε?
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Erdős Problem #141
Let k≥3. Are there k consecutive primes in arithmetic progression?
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Erdős Problem #142
Prove an asymptotic formula for r_k(N), the largest possible size of a subset of 1, …, N that does not contain any non-trivial k-term arithmetic progression. That is, find f_k with r_k(N) / f_k(N) → 1 as N → ∞.
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Erdős Problem #143
Does this imply that liminf |A ∩ [1,x]|/x = 0?
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Erdős Problem #145
Let s_1 < s_2 < ⋯ be the sequence of squarefree numbers. Is it true that, for any α≥ 0, lim_x→∞ 1/xΣ_s_n≤ x(s_n+1-s_n)^α exists?
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Erdős Problem #148
Let F(k) be the number of solutions to 1= 1/n_1+⋯+1/n_k, where 1≤ n_1<⋯<n_k are distinct integers. Find good estimates for F(k).
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Erdős Problem #15
Is it true that Σ_n=1^∞(-1)^nn/p_n converges, where p_n is the sequence of primes? Note: In the problem statement, p_n is the n-th prime, indexed such that p_1=2, p_2=3, …. We 0-index here to reflect how Nat.nth works.
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Erdős Problem #153
Let A be a finite Sidon set and A+A=s_1<⋯<s_t. Is it true that 1/tΣ_1≤ i<t(s_i+1-s_i)^2 → ∞ as lvert Arvert→ ∞?
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Erdős Problem #155
Is it true that for every k ≥ 1 we have F(N + k) ≤ F(N) + 1 for all sufficiently large N?
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Erdős Problem #156
Does there exist a maximal Sidon set A⊂ 1,…,N of size O(N^1/3)? A question of Erdős, Sárközy, and Sós [ESS94].
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Erdős Problem #158
Let A be an infinite B₂[2] set. Must liminf |A ∩ 1, ..., N| * N ^ (- 1 / 2) = 0?
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Erdős Problem #159
There exists some constant c>0 such that R(C_4,K_n) ≪ n^2-c. The prize of 100 is offered in [Er78] for a proof or disproof. This problem is #17 in Ramsey Theory in the graphs problem collection.
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Erdős Problem #160
Estimate h(n) by finding a better upper bound.
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Erdős Problem #168
What is the limit F(N)/N as N → ∞?
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Erdős Problem #17
Erdős Problem 17. Are there infinitely many cluster primes?
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Erdős Problem #170
The problem is to determine the limit of the sequence F(N)/√(N) as N → ∞.
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Erdős Problem #172
Is it true that in any finite colouring of ℕ there exist arbitrarily large finite A such that all sums and products of distinct elements in A are the same colour?
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Erdős Problem #18
Conjecture 1. Are there infinitely many practical numbers m such that h(m) < (log log m)^O(1)? More precisely: does there exist a constant C > 0 such that for infinitely many practical numbers m, we have h(m) < (log log m)^C?
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Erdős Problem #181
Let Q_n be the n-dimensional hypercube graph (so that Q_n has 2^n vertices and n2^n-1 edges). Prove that R(Q_n) ≪ 2^n.
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Erdős Problem #184
Any graph on n vertices can be decomposed into O(n) many edge-disjoint cycles and edges.
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Erdős Problem #188
What is the smallest k such that ℝ^2 can be red/blue coloured with no pair of red points unit distance apart, and no k-term arithmetic progression of blue points with distance 1?
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Erdős Problem #19
If G is an edge-disjoint union of n copies of K_n, then is χ(G) = n?
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Erdős Problem #195
What is the largest k such that in any permutation of ℤ there must exist a monotone k-term arithmetic progression x_1 < ⋯ < x_k? Here a permutation of ℤ is a one-sided arrangement a_1, a_2, a_3, … of the integers, i.e.
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Erdős Problem #196
Must every permutation of ℕ, contain a monotone 4-term arithmetic progression?
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Erdős Problem #197
Can ℕ be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithmetic progressions?
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Erdős Problem #200
Does the longest arithmetic progression of primes in 1,…,N have length o(log N)?
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Erdős Problem #203
Is there an integer m with (m, 6) = 1 such that none of 2^k · 3^ℓ · m + 1 are prime, for any k, ℓ ≥ 0?
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Erdős Problem #208
Let s_1 < s_2 < … be the sequence of squarefree numbers. Is it true that for any ε > 0 and large n, s_n+1 - s_n ≪_ε s_n^ε?
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Erdős Problem #212
Is there a dense subset of ℝ^2 such that all pairwise distances are rational?
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Erdős Problem #213
Let n ≥ 4. Are there n points in ℝ^2, no three on a line and no four on a circle, such that all pairwise distances are integers?
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Erdős Problem #23
Can every triangle-free graph on 5n vertices be made bipartite by deleting at most n^2 edges?
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Erdős Problem #233
A conjecture by Heath-Brown: The sum of squares of the first N gaps between consecutive primes behaves like N * (log N)^2.
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Erdős Problem #234
Is it true that for all c ≥ 0, the density f c of integers for which (p (n + 1) - p n) / log n < c exists and is a continuous function of c?
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Erdős Problem #236
Let f(n) count the number of solutions to n=p+2^k for prime p and k≥ 0. Show that f(n)=o(log n).
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Erdős Problem #238
Let c₁, c₂ > 0. Is it true that for any sufficiently large x, there exists more than c₁ * log x many consecutive primes ≤ x such that the difference between any two is > c₂?
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Erdős Problem #241
Is it true that f(N)∼ N^1/3? Originally asked to Erdős by Bose. This is discussed in problem C11 of Guy's collection [Gu04].
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Erdős Problem #243
Let a_1 < a_2 < … be a sequence of integers such that lim_n→∞ a_n/a_n-1^2 = 1 and Σ 1/a_n ∈ ℚ. Then, for all sufficiently large n ≥ 1, a_n = a_n-1^2 - a_n-1 + 1.
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Erdős Problem #244
Let C > 1. Does the set of integers of the form p + ⌊ C^k ⌋, for some prime p and k≥ 0, have density >0?
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Erdős Problem #247
Let n_1 < n_2 < ⋯ be a sequence of integers such that limsup n_k/k = ∞. Is Σ_k=1^∞ 1/2^n_k transcendental?
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Erdős Problem #249
Is Σ_n φ(n)/2^n irrational? Here φ is the Euler totient function.
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Erdős Problem #25
Let n_1 < n_2 < … be an arbitrary sequence of integers, each with an associated residue class a_i pmodn_i. Let A be the set of integers n such that for every i either n < n_i or n not≡ a_i pmodn_i. Must the logarithmic density of A exist?
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Erdős Problem #251
Is Σ_n=1^∞ p_n/2^n irrational? Here p_n is the n-th prime (p_1=2, p_2=3, …).
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Erdős Problem #252
Erdős Problem 252: irrationality of the sum for a given k.
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Erdős Problem #257
Let A⊆ℕ be an infinite set. Is Σ_n∈ A 1/2^n - 1 irrational?
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Erdős Problem #261
Do all positive integers n have the required property?
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Erdős Problem #263
Is a_n = 2^2^n an irrationality sequence in the above sense?
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Erdős Problem #264
Is n! an example of an irrationality sequence?
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Erdős Problem #272
Let N≥ 1. What is the largest t such that there are A_1,…,A_t⊆ 1,…,N with A_i∩ A_j a non-empty arithmetic progression for all i≠ j?
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Erdős Problem #273
Is there a covering system all of whose moduli are of the form p-1 for some primes p ≥ 5?
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