Erdős Problems

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

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576 Problems · 1 with reviewed evidence

4/12

Number	Question	Source says	Formal	Result here
#322	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#323	Is it true that $f_{k,k}(x) \gg_\epsilon x^{1-\epsilon}$ for all $\epsilon>0$ ?openFormalized	open	In Lean	No Result decision
#324	Does there exist a polynomial $f(x)\in\mathbb{Z}[x]$ such that all the sums $f(a)+f(b)$ with $a < b$ nonnegative integers are distinct?openFormalized	open	In Lean	No Result decision
#325	Writing $f_{k, 3}(x)$ for the number of integers $\leq x$ which are the sum of three $k$ th powers, is it true that $f_{k, 3}(x) \gg x ^ (3 / k)$ ?openFormalized	open	In Lean	No Result decision
#326	Let $A \subset \mathbb{N}$ be an additive basis of order 2.openFormalized	open	In Lean	No Result decision
#327	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#328	Suppose $A\subseteq\mathbb{N}$ and $C>0$ is such that $1_A\ast 1_A(n)\leq C$ for all $n\in\mathbb{N}$ . Can $A$ be partitioned into $t$ many subsets $A_1,\ldots,A_t$ (where $t=t(C)$ depends only on $C$ ) such that $1_{A_i}\ast 1_{A_i}(n)<C$ for all $1\leq i\leq t$ and $n\in \mathbb{N}$ ?disproved (Lean)Formalized	disproved (Lean)	In Lean	No Result decision
#329	Erdős Problem 329. Let `A ⊆ ℕ` be a Sidon set. How large can `lim sup_{N → ∞} \|A ∩ {1,…,N}\| / N^{1/2}` be?openFormalized	open	In Lean	No Result decision
#330	Does there exist a minimal basis $A \subset \mathbb{N}$ with positive density such that, for any $n \in A$ , the (upper) density of integers which cannot be represented without using $n$ is positive?proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#331	Let $A,B\subseteq \mathbb{N}$ such that for all large $N$ $\lvert A\cap \{1,\ldots,N\}\rvert \gg N^{1/2}$ and $\lvert B\cap \{1,\ldots,N\}\rvert \gg N^{1/2}.$ Is it true that there are infinitely many solutions to $a_1-a_2=b_1-b_2\neq 0$ with $a_1,a_2\in A$ and $b_1,b_2\in B$ ?disproved (Lean)Formalized	disproved (Lean)	In Lean	No Result decision
#332	Let $A\subseteq \mathbb{N}$ and $D(A)$ be the set of those numbers which occur infinitely often as $a_1 - a_2$ with $a_1, a_2\in A$ . What conditions on $A$ are sufficient to ensure $D(A)$ has bounded gaps?openFormalized	open	In Lean	No Result decision
#333	Let $A\subseteq \mathbb{N}$ be a set of density zero. Does there exist a $B$ such that $A\subseteq B+B$ and $\lvert B\cap \{1,\ldots,N\}\rvert =o(N^{1/2})$ for all large $N$ ?disproved (Lean)Formalized	disproved (Lean)	In Lean	No Result decision
#334	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#335	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#336	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#337	Let $A\subseteq \mathbb{N}$ be an additive basis (of any finite order) such that $\lvert A\cap \{1,\ldots,N\}\rvert=o(N)$ . Is it true that $\lim_{N\to \infty}\frac{\lvert (A+A)\cap \{1,\ldots,N\}\rvert} {\lvert A\cap \{1,\ldots,N\}\rvert}=\infty?$ disproved (Lean)Formalized	disproved (Lean)	In Lean	No Result decision
#338	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#339	No statement retained — open to read what the source holdsprovedNo formal declaration	proved	No formal statement	No Result decision
#340	Let $A = \{1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, \ldots\}$ be the greedy Sidon sequence: we begin with $1$ and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no non-trivial solutions to $a + b = c + d$ ). What is the order of growth of $A$ ? Is it true that $\|A \cap \{1, \ldots, N\}\| \gg N^{1/2 - \varepsilon}$ for all $\varepsilon > 0$ and large $N$ ?openFormalized	open	In Lean	No Result decision
#341	Let $A=\{a_1 < \cdots < a_k\}$ be a finite set of integers and extend it to an infinite sequence $\overline{A}=\{a_1 < a_2 < \cdots \}$ by defining $a_{n+1}$ for $n \geq k$ to be the least integer exceeding $a_n$ which is not of the form $a_i + a_j$ with $i,j \leq n$ . Is it true that the sequence of differences $a_{m+1}-a_m$ is eventually periodic?openFormalized	open	In Lean	No Result decision
#342	Do infinitely many pairs $(a, a+2)$ occur in Ulam's sequence?openFormalized	open	In Lean	No Result decision
#343	No statement retained — open to read what the source holdsprovedNo formal declaration	proved	No formal statement	No Result decision
#344	No statement retained — open to read what the source holdsprovedNo formal declaration	proved	No formal statement	No Result decision
#345	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#346	Is 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`?openFormalized	open	In Lean	No Result decision
#347	Is there a sequence $A=\{a_1\leq a_2\leq \cdots\}$ of integers with $\lim \frac{a_{n+1}}{a_n}=2$ such that $P(A')= \left\{\sum_{n\in B}n : B\subseteq A'\textrm{ finite }\right\}$ has density $1$ for every cofinite subsequence $A'$ of $A$ ?proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#348	For what values of $0 \leq m < n$ is there a complete sequence $A = \{a_1 \leq a_2 \leq \cdots\}$ of integers such that 1. $A$ remains complete after removing any $m$ elements, but 2. $A$ is not complete after removing any $n$ elements.openFormalized	open	In Lean	No Result decision
#349	For $\alpha > 2$ and any $t > 0$ , the sequence $\lfloor t\alpha^n\rfloor$ is not additively complete; equivalently $(t, \alpha)$ is not a "good pair". A partial result on the open Erdős Problem 349: it complements `complete_for_alpha_in_Ioo_one_to_goldenRatio`.openFormalized	open	In Lean	No Result decision
#350	Small sanity check: the two predicates are saying the same thing.proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#351	Let $p(x) \in \mathbb{Q}[x]$ be a non-constant rational polynomial with positive leading coefficient. Is it true that $A=\{ p(n)+1/n : n \in \mathbb{N}\}$ is strongly complete, in the sense that, for any finite set $B$ , $\left\{\sum_{a \in X} a : X \subseteq A \setminus B, X \textrm{ is finite}\right\}$ contains all sufficiently large integers?proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#354	Let $\alpha,\beta\in \mathbb{R}_{>0}$ such that $\alpha/\beta$ is irrational. Is $\{ \lfloor \alpha\rfloor,\lfloor \gamma\alpha\rfloor,\lfloor \gamma^2\alpha\rfloor,\ldots\}\cup \{ \lfloor \beta\rfloor,\lfloor \gamma\beta\rfloor,\lfloor \gamma^2\beta\rfloor,\ldots\}$ complete?openFormalized	open	In Lean	No Result decision
#355	Is there a lacunary sequence $A\subseteq \mathbb{N}$ (so that $A=\{a_1 < \cdots\}$ and there exists some $\lambda > 1$ such that $a_{n+1}/a_n\geq \lambda$ for all $n\geq 1$ ) such that $\left\{ \sum_{a\in A'}\frac{1}{a} : A'\subseteq A\textrm{ finite}\right\}$ contain all rationals in some open interval?proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#356	No statement retained — open to read what the source holdsprovedNo formal declaration	proved	No formal statement	No Result decision
#357	Let $f(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 < \dotsc < a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct. Is $f(n)=o(n)$ ?openFormalized	open	In Lean	No Result decision
#358	When $A_n = n$ , the function $f$ defined above counts the number of odd divisors of $n$ .provedFormalized	proved	In Lean	No Result decision
#359	Let $a_1< a_2 < ⋯$ be an infinite sequence of integers such that $a_1=1$ and $a_{i+1}$ is the least integer which is not a sum of consecutive earlier $a_j$ s. Show that $a_k / k \to \infty$ .openFormalized	open	In Lean	No Result decision
#360	No statement retained — open to read what the source holdssolvedNo formal declaration	solved	No formal statement	No Result decision
#361	Let $c > 0$ and $n$ be some large integer. What is the size of the largest set $A \subseteq \{1, \ldots, \lfloor c n \rfloor\}$ such that $n$ is not a sum of a subset of $A$ ? Does this depend on $n$ in an irregular way?openFormalized	open	In Lean	No Result decision
#362	No statement retained — open to read what the source holdsprovedNo formal declaration	proved	No formal statement	No Result decision
#363	Is it true that there are only finitely many collections of disjoint intervals $I_1,\ldots,I_n$ of size $\lvert I_i\rvert \geq 4$ for $1\leq i\leq n$ such that $\prod_{1\leq i\leq n}\prod_{m\in I_i}m$ is a square?disproved (Lean)Formalized	disproved (Lean)	In Lean	No Result decision
#364	There is no consecutive triple of powerful numbers.verifiableFormalized	verifiable	In Lean	No Result decision
#365	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#366	Are there any $2$ -full $n$ such that $n+1$ is $3$ -full?verifiableFormalized	verifiable	In Lean	No Result decision
#367	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#368	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#369	Let $\epsilon>0$ and $k\geq 2$ . Is it true that, for all sufficiently large $n$ , there is a sequence of $k$ consecutive integers in $\{1,\ldots,n\}$ all of which are $n^\epsilon$ -smooth?proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#370	Are there infinitely many $n$ such that the largest prime factor of $n$ is $< n^{\frac{1}{2}}$ and the largest prime factor of $n + 1$ is $< (n + 1)^{\frac{1}{2}}$ .proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#371	Let $P(n)$ denote the largest prime factor of $n$ . Show that the set of $n$ with $P(n+1) > P(n)$ has density $\frac{1}{2}$ .openFormalized	open	In Lean	No Result decision

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