Skip to content

Erdős Problems

1,217 source-owned questions · 604 with a formal statement · searchable by statement, number, topic and source status.

Add a contributionCollection details
Full distribution

Collection coverage

Source status, exact formal material, and reviewed Results are separate signals.

  • Open per source608
  • Resolved per source556
  • Other source status53
Exact formal statement available604 / 1,217
With Repository-reviewed evidence2 / 1,217
More filters

Coverage is source-observation coverage, not Problem completeness. Inspect coverage

Problems

576 Problems · 1 with reviewed evidence

4/12

SelectNumberQuestionOpen
#322No statement retained — open to read what the source holdsSource says openNo formal declaration
#323Is it true that fk,k(x)≫ϵx1−ϵf_{k,k}(x) \gg_\epsilon x^{1-\epsilon} for all ϵ>0\epsilon>0?Source says openFormalized
#324Does there exist a polynomial f(x)∈Z[x]f(x)\in\mathbb{Z}[x] such that all the sums f(a)+f(b)f(a)+f(b) with a<ba < b nonnegative integers are distinct?Source says openFormalized
#325Writing fk,3(x)f_{k, 3}(x) for the number of integers ≤x\leq x which are the sum of three kkth powers, is it true that fk,3(x)≫x(3/k)f_{k, 3}(x) \gg x ^ (3 / k)?Source says openFormalized
#326Let A⊂NA \subset \mathbb{N} be an additive basis of order 2.Source says openFormalized
#327No statement retained — open to read what the source holdsSource says openNo formal declaration
#328Suppose A⊆NA\subseteq\mathbb{N} and C>0C>0 is such that 1A∗1A(n)≤C1_A\ast 1_A(n)\leq C for all n∈Nn\in\mathbb{N}. Can AA be partitioned into tt many subsets A1,…,AtA_1,\ldots,A_t (where t=t(C)t=t(C) depends only on CC) such that 1Ai∗1Ai(n)<C1_{A_i}\ast 1_{A_i}(n)<C for all 1≤i≤t1\leq i\leq t and n∈Nn\in \mathbb{N}?Source says disproved (Lean)Formalized
#329Erdős Problem 329. Let A ⊆ ℕ be a Sidon set. How large can lim sup_{N → ∞} |A ∩ {1,…,N}| / N^{1/2} be?Source says openFormalized
#330Does there exist a minimal basis A⊂NA \subset \mathbb{N} with positive density such that, for any n∈An \in A, the (upper) density of integers which cannot be represented without using nn is positive?Source says proved (Lean)Formalized
#331Let A,B⊆NA,B\subseteq \mathbb{N} such that for all large NN∣A∩{1,…,N}∣≫N1/2\lvert A\cap \{1,\ldots,N\}\rvert \gg N^{1/2}and∣B∩{1,…,N}∣≫N1/2.\lvert B\cap \{1,\ldots,N\}\rvert \gg N^{1/2}. Is it true that there are infinitely many solutions to a1−a2=b1−b2≠0a_1-a_2=b_1-b_2\neq 0 with a1,a2∈Aa_1,a_2\in A and b1,b2∈Bb_1,b_2\in B?Source says disproved (Lean)Formalized
#332Let A⊆NA\subseteq \mathbb{N} and D(A)D(A) be the set of those numbers which occur infinitely often as a1−a2a_1 - a_2 with a1,a2∈Aa_1, a_2\in A. What conditions on AA are sufficient to ensure D(A)D(A) has bounded gaps?Source says openFormalized
#333Let A⊆NA\subseteq \mathbb{N} be a set of density zero. Does there exist a BB such that A⊆B+BA\subseteq B+B and ∣B∩{1,…,N}∣=o(N1/2)\lvert B\cap \{1,\ldots,N\}\rvert =o(N^{1/2}) for all large NN?Source says disproved (Lean)Formalized
#334No statement retained — open to read what the source holdsSource says openNo formal declaration
#335No statement retained — open to read what the source holdsSource says openNo formal declaration
#336No statement retained — open to read what the source holdsSource says openNo formal declaration
#337Let A⊆NA\subseteq \mathbb{N} be an additive basis (of any finite order) such that ∣A∩{1,…,N}∣=o(N)\lvert A\cap \{1,\ldots,N\}\rvert=o(N). Is it true that lim⁡N→∞∣(A+A)∩{1,…,N}∣∣A∩{1,…,N}∣=∞? \lim_{N\to \infty}\frac{\lvert (A+A)\cap \{1,\ldots,N\}\rvert} {\lvert A\cap \{1,\ldots,N\}\rvert}=\infty? Source says disproved (Lean)Formalized
#338No statement retained — open to read what the source holdsSource says openNo formal declaration
#339No statement retained — open to read what the source holdsSource says provedNo formal declaration
#340Let A={1,2,4,8,13,21,31,45,66,81,97,…}A = \{1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, \ldots\} be the greedy Sidon sequence: we begin with 11 and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no non-trivial solutions to a+b=c+da + b = c + d). What is the order of growth of AA? Is it true that ∣A∩{1,…,N}∣≫N1/2−ε|A \cap \{1, \ldots, N\}| \gg N^{1/2 - \varepsilon} for all ε>0\varepsilon > 0 and large NN?Source says openFormalized
#341Let A={a1<⋯<ak}A=\{a_1 < \cdots < a_k\} be a finite set of integers and extend it to an infinite sequence A‾={a1<a2<⋯ }\overline{A}=\{a_1 < a_2 < \cdots \} by defining an+1a_{n+1} for n≥kn \geq k to be the least integer exceeding ana_n which is not of the form ai+aja_i + a_j with i,j≤ni,j \leq n. Is it true that the sequence of differences am+1−ama_{m+1}-a_m is eventually periodic?Source says openFormalized
#342Do infinitely many pairs (a,a+2)(a, a+2) occur in Ulam's sequence?Source says openFormalized
#343No statement retained — open to read what the source holdsSource says provedNo formal declaration
#344No statement retained — open to read what the source holdsSource says provedNo formal declaration
#345No statement retained — open to read what the source holdsSource says openNo formal declaration
#346Is it true that for every lacunary, strongly complete sequence A that is not complete whenever infinitely many terms are removed from it, lim A (n + 1) / A n = (1 + √5) / 2?Source says openFormalized
#347Is there a sequence A={a1≤a2≤⋯ }A=\{a_1\leq a_2\leq \cdots\} of integers with lim⁡an+1an=2\lim \frac{a_{n+1}}{a_n}=2 such that P(A′)={∑n∈Bn:B⊆A′ finite }P(A')= \left\{\sum_{n\in B}n : B\subseteq A'\textrm{ finite }\right\} has density 11 for every cofinite subsequence A′A' of AA?Source says proved (Lean)Formalized
#348For what values of 0≤m<n0 \leq m < n is there a complete sequence A={a1≤a2≤⋯ }A = \{a_1 \leq a_2 \leq \cdots\} of integers such that 1. AA remains complete after removing any mm elements, but 2. AA is not complete after removing any nn elements.Source says openFormalized
#349For α>2\alpha > 2 and any t>0t > 0, the sequence ⌊tαn⌋\lfloor t\alpha^n\rfloor is not additively complete; equivalently (t,α)(t, \alpha) is not a "good pair". A partial result on the open Erdős Problem 349: it complements complete_for_alpha_in_Ioo_one_to_goldenRatio.Source says openFormalized
#350Small sanity check: the two predicates are saying the same thing.Source says proved (Lean)Formalized
#351Let p(x)∈Q[x]p(x) \in \mathbb{Q}[x] be a non-constant rational polynomial with positive leading coefficient. Is it true that A={p(n)+1/n:n∈N}A=\{ p(n)+1/n : n \in \mathbb{N}\} is strongly complete, in the sense that, for any finite set BB, {∑a∈Xa:X⊆A∖B,X is finite}\left\{\sum_{a \in X} a : X \subseteq A \setminus B, X \textrm{ is finite}\right\} contains all sufficiently large integers?Source says proved (Lean)Formalized
#354Let α,β∈R>0\alpha,\beta\in \mathbb{R}_{>0} such that α/β\alpha/\beta is irrational. Is {⌊α⌋,⌊γα⌋,⌊γ2α⌋,…}∪{⌊β⌋,⌊γβ⌋,⌊γ2β⌋,…}\{ \lfloor \alpha\rfloor,\lfloor \gamma\alpha\rfloor,\lfloor \gamma^2\alpha\rfloor,\ldots\}\cup \{ \lfloor \beta\rfloor,\lfloor \gamma\beta\rfloor,\lfloor \gamma^2\beta\rfloor,\ldots\} complete?Source says openFormalized
#355Is there a lacunary sequence A⊆NA\subseteq \mathbb{N} (so that A={a1<⋯ }A=\{a_1 < \cdots\} and there exists some λ>1\lambda > 1 such that an+1/an≥λa_{n+1}/a_n\geq \lambda for all n≥1n\geq 1) such that {∑a∈A′1a:A′⊆A finite}\left\{ \sum_{a\in A'}\frac{1}{a} : A'\subseteq A\textrm{ finite}\right\} contain all rationals in some open interval?Source says proved (Lean)Formalized
#356No statement retained — open to read what the source holdsSource says provedNo formal declaration
#357Let f(n)f(n) be the maximal kk such that there exist integers 1≤a1<…<ak≤n1 \le a_1 < \dotsc < a_k \le n such that all sums of the shape ∑u≤i≤vai\sum_{u \le i \le v} a_i are distinct. Is f(n)=o(n)f(n)=o(n)?Source says openFormalized
#358When An=nA_n = n, the function ff defined above counts the number of odd divisors of nn.Source says provedFormalized
#359Let a1<a2<⋯a_1< a_2 < ⋯ be an infinite sequence of integers such that a1=1a_1=1 and ai+1a_{i+1} is the least integer which is not a sum of consecutive earlier aja_js. Show that ak/k→∞a_k / k \to \infty.Source says openFormalized
#360No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#361Let c>0c > 0 and nn be some large integer. What is the size of the largest set A⊆{1,…,⌊cn⌋}A \subseteq \{1, \ldots, \lfloor c n \rfloor\} such that nn is not a sum of a subset of AA? Does this depend on nn in an irregular way?Source says openFormalized
#362No statement retained — open to read what the source holdsSource says provedNo formal declaration
#363Is it true that there are only finitely many collections of disjoint intervals I1,…,InI_1,\ldots,I_n of size ∣Ii∣≥4\lvert I_i\rvert \geq 4 for 1≤i≤n1\leq i\leq n such that∏1≤i≤n∏m∈Iim\prod_{1\leq i\leq n}\prod_{m\in I_i}mis a square?Source says disproved (Lean)Formalized
#364There is no consecutive triple of powerful numbers.Source says verifiableFormalized
#365No statement retained — open to read what the source holdsSource says openNo formal declaration
#366Are there any 22-full nn such that n+1n+1 is 33-full?Source says verifiableFormalized
#367No statement retained — open to read what the source holdsSource says openNo formal declaration
#368No statement retained — open to read what the source holdsSource says openNo formal declaration
#369Let ϵ>0\epsilon>0 and k≥2k\geq 2. Is it true that, for all sufficiently large nn, there is a sequence of kk consecutive integers in {1,…,n}\{1,\ldots,n\} all of which are nϵn^\epsilon-smooth?Source says proved (Lean)Formalized
#370Are there infinitely many nn such that the largest prime factor of nn is <n12< n^{\frac{1}{2}} and the largest prime factor of n+1n + 1 is <(n+1)12< (n + 1)^{\frac{1}{2}}.Source says proved (Lean)Formalized
#371Let P(n)P(n) denote the largest prime factor of nn. Show that the set of nn with P(n+1)>P(n)P(n+1) > P(n) has density 12\frac{1}{2}.Source says openFormalized