Erdős Problems

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

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With Repository-reviewed evidence2 / 1,217

576 Problems · 1 with reviewed evidence

9/12

Number	Question	Source says	Formal	Result here
#773	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#774	Is every proportionately dissociated (infinite) set the union of a finite number of dissociated sets?openFormalized	open	In Lean	No Result decision
#779	A Conjecture of Marian Deaconescu, see p.120 in https://doi.org/10.2307/2975810falsifiableFormalized	falsifiable	In Lean	No Result decision
#782	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#783	No statement retained — open to read what the source holdssolvedNo formal declaration	solved	No formal statement	No Result decision
#784	No statement retained — open to read what the source holdssolvedNo formal declaration	solved	No formal statement	No Result decision
#786	Let $\epsilon > 0$ . Is there some set $A\subset\mathbb{N}$ of density $> 1 - \epsilon$ such that $a_1\cdots a_r = b_1\cdots b_s$ with $a_i, b_j\in A$ can only hold when $r = s$ ?openFormalized	open	In Lean	No Result decision
#793	No statement retained — open to read what the source holdsproved (Lean)No formal declaration	proved (Lean)	No formal statement	No Result decision
#795	No statement retained — open to read what the source holdsprovedNo formal declaration	proved	No formal statement	No Result decision
#796	Let $k\geq 2$ and let $g_k(n)$ be the largest possible size of $A\subseteq \{1,\ldots,n\}$ such that every $m$ has $<k$ solutions to $m=a_1a_2$ with $a_1<a_2\in A$ . Is it true that $g_3(n)=\frac{\log\log n}{\log n}n+(c+o(1))\frac{n}{\log n}$ for some constant $c$ ?openFormalized	open	In Lean	No Result decision
#820	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#821	Is it true that, for every $\epsilon>0$ , there exist infinitely many $n$ such that $g(n) > n^{1-\epsilon}$ ?openFormalized	open	In Lean	No Result decision
#822	Does the set of integers of the form $n + \varphi(n)$ have positive (lower) density?provedFormalized	proved	In Lean	No Result decision
#823	No statement retained — open to read what the source holdsprovedNo formal declaration	proved	No formal statement	No Result decision
#824	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#825	Is there an absolute constant $C > 0$ such that every integer $n$ with $\sigma(n) > Cn$ is the distinct sum of proper divisors of $n$ ?provedFormalized	proved	In Lean	No Result decision
#826	Are there infinitely many $n$ such that, for all $k\geq 1$ $\tau(n + k) \ll k?$ openFormalized	open	In Lean	No Result decision
#828	Is it true that, for any $a \in \mathbb{Z}$ , there are infinitely many $n$ such that $\phi(n) \| n + a$ ?openFormalized	open	In Lean	No Result decision
#829	Erdős Problem 829 (open). Let $A \subseteq \mathbb{N}$ be the set of perfect cubes. Is it true that $(1_A \ast 1_A)(n) \ll (\log n)^{O(1)}$ ? 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 $n \to \infty$ ?openFormalized	open	In Lean	No Result decision
#830	Erdos Problem 830, Part 1 We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$ . Are there infinitely many amicable pairs?openFormalized	open	In Lean	No Result decision
#839	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#841	No statement retained — open to read what the source holdssolvedNo formal declaration	solved	No formal statement	No Result decision
#843	No statement retained — open to read what the source holdsprovedNo formal declaration	proved	No formal statement	No Result decision
#844	Let $A\subseteq \{1,\ldots,N\}$ be such that, for all $a,b\in A$ , the product $ab$ is not squarefree.proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#845	Let $C > 0$ . Is it true that the set of integers of the form $n = b_1 + \cdots + b_t$ , with $b_1 < \cdots < b_t$ , where $b_i = 2^{k_i}3^{l_i}$ for $1 \leq i\leq t$ and $b_t \leq Cb_1$ has density $0$ ?disproved (Lean)Formalized	disproved (Lean)	In Lean	No Result decision
#848	Is the maximum size of a set $A ⊆ \{1, \dots, N\}$ such that $ab + 1$ is never squarefree (for all $a, b ∈ A$ ) achieved by taking those $n ≡ 7 \pmod{25}$ ?decidableFormalized	decidable	In Lean	No Result decision
#849	Is it true that, for every integer $t\geq1$ , there is some integer $a$ such that ${n \choose k} = a$ with $1\leq k \le \frac{n}{2}$ has exactly $t$ solutions?openFormalized	open	In Lean	No Result decision
#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?openFormalized	open	In Lean	No Result decision
#851	Let $\epsilon > 0$ . Is there some $r \ll_\epsilon 1$ such that the density of integers of the form $2^k+n$ , where $k \geq 0$ and $n$ has at most $r$ prime divisors, is at least $1-\epsilon$ ?provedFormalized	proved	In Lean	No Result decision
#852	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#853	Let $d_n = p_{n+1} - p_n$ , where $p_n$ is the $n$ th prime. Let $r(x)$ be the smallest even integer $t$ such that $d_n = t$ has no solutions for $n \le x$ .openFormalized	open	In Lean	No Result decision
#854	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#855	Erdős Problem 855 (Segal's conjecture): $\pi(x + y) \le \pi(x) + \pi(y)$ for sufficiently large $x, y$ .openFormalized	open	In Lean	No Result decision
#856	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#858	No statement retained — open to read what the source holdssolvedNo formal declaration	solved	No formal statement	No Result decision
#859	The density of the divisor sum set is asymptotically equivalent to $c_1 / \log(t)^{c_2}$ .openFormalized	open	In Lean	No Result decision
#860	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#861	No statement retained — open to read what the source holdssolvedNo formal declaration	solved	No formal statement	No Result decision
#862	Let $A_1(N)$ be the number of maximal Sidon subsets of $\{1,\ldots,N\}$ . Is it true that $A_1(N) < 2^{o(N^{1/2})}?$ solved (Lean)Formalized	solved (Lean)	In Lean	No Result decision
#863	No statement retained — open to read what the source holdsprovedNo formal declaration	proved	No formal statement	No Result decision
#864	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#865	There exists a constant $C>0$ such that, for all large $N$ , if $A\subseteq \{1,\ldots,N\}$ has size at least $\frac{5}{8}N+C$ then there are distinct $a,b,c\in A$ such that $a+b,a+c,b+c\in A$ .proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#866	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#868	Let $A$ be an additive basis of order $2$ , let $f(n)$ denote the number of ways in which $n$ can be written as the sum of two elements from $A$ . If $f(n) \to \infty$ as $n \to \infty$ , then must $A$ contain a minimal additive basis of order $2$ ?solvedFormalized	solved	In Lean	No Result decision
#869	No statement retained — open to read what the source holdsdisprovedNo formal declaration	disproved	No formal statement	No Result decision
#870	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#871	Let $A$ be an additive basis of order $2$ , and suppose $1_A\ast 1_A(n)\to \infty$ as $n\to \infty$ . Can $A$ be partitioned into two disjoint additive bases of order $2$ ?disproved (Lean)Formalized	disproved (Lean)	In Lean	No Result decision
#872	Each move claims exactly one pool element, so the minimax value never exceeds the number of already claimed elements plus the number of still unclaimed elements.openFormalized	open	In Lean	No Result decision

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