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Erdős Problems

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

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1,217 Problems · 2 with reviewed evidence

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#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
#352Is there some c>0c > 0 such that every measurable A⊆R2A \subseteq \mathbb{R}^2 of measure ≥c\geq c contains the vertices of a triangle of area 1?Source says openFormalized
#353Let A⊆R2A\subseteq \mathbb{R}^2 be a measurable set with infinite measure. Must AA contain the vertices of an isosceles trapezoid of area 11? What about an isosceles triangle, or a right-angled triangle, or a cyclic quadrilateral, or a convex polygon with congruent sides?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
#372No statement retained — open to read what the source holdsSource says provedNo formal declaration
#373Show that the equation n!=a_1!a_2!···a_k!, with n−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k, has only finitely many solutions.Source says openFormalized
#374No statement retained — open to read what the source holdsSource says openNo formal declaration
#375Is Erdos375Prop true?Source says falsifiableFormalized
#376Are there infinitely many nn such that (2nn){2n\choose n} is coprime to 105105?Source says openFormalized
#377Is there some absolute constant C>0C > 0 such that ∑p≤n1p∤(2nn)1p≤C \sum_{p \leq n} 1_{p\nmid {2n \choose n}}\frac{1}{p} \leq C for all nn?Source says openFormalized
#378No statement retained — open to read what the source holdsSource says provedNo formal declaration
#379Let S(n)S(n) denote the largest integer such that, for all 1≤k<n1 ≤ k < n, the binomial coefficient (nk)\binom{n}{k} is divisible by pS(n)p^S(n) for some prime pp (depending on kk).Then lim sup⁡S(n)=∞\limsup S(n) = \infty.Source says proved (Lean)Formalized
#380No statement retained — open to read what the source holdsSource says provedNo formal declaration
#381No statement retained — open to read what the source holdsSource says disprovedNo formal declaration
#382No statement retained — open to read what the source holdsSource says openNo formal declaration
#383Is it true that for every kk there are infinitely many primes pp such that the largest prime divisor of ∏i=0k(p2+i) \prod_{i = 0}^k (p ^ 2 + i) is pp?Source says openFormalized
#384No statement retained — open to read what the source holdsSource says provedNo formal declaration