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

18/26

Number	Question	Source says	Formal	Result here
#817	Let $k \geq 3$ . Define $g_k(n)$ to be the minimal $N$ such that $\{1, ..., N\}$ contains some $A$ of size $\|A\| = n$ such that $\langle A\rangle = \left\{\sum_{a \in A} \epsilon_a a : \epsilon_a \in\{0, 1\}\right\}$ contains no non-trivial $k$ -term arithmetic progression. Estimate $g_k(n)$ . In particular, is it true that $g_3(n) \gg 3^n$ openFormalized	open	In Lean	No Result decision
#818	Let $A$ be a finite set of integers such that $\lvert A+A\rvert \ll \lvert A\rvert$ . Is it true that $\lvert AA\rvert \gg \frac{\lvert A\rvert^2}{(\log \lvert A\rvert)^C}$ for some constant $C>0$ ?proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#819	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	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
#827	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	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
#831	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#832	No statement retained — open to read what the source holdsdisprovedNo formal declaration	disproved	No formal statement	No Result decision
#833	No statement retained — open to read what the source holdsprovedNo formal declaration	proved	No formal statement	No Result decision
#834	No statement retained — open to read what the source holdssolvedNo formal declaration	solved	No formal statement	No Result decision
#835	It is known that for $3 \leq k \leq 8$ , the chromatic number of $J(2k, k)$ is greater than $k+1$ , see [Johnson graphs](https://aeb.win.tue.nl/graphs/Johnson.html).verifiableFormalized	verifiable	In Lean	No Result decision
#836	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#837	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#838	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#839	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#840	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
#842	No statement retained — open to read what the source holdsprovedNo formal declaration	proved	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
#846	Erdős Problem 846 Let `A ⊂ ℝ²` be an infinite set for which there exists some `ϵ>0` such that in any subset of `A` of size `n` there are always at least `ϵn` with no three on a line. Is it true that `A` is the union of a finite number of sets where no three are on a line?disproved (Lean)Formalized	disproved (Lean)	In Lean	No Result decision
#847	Let $A \subset \mathbb{N}$ be an infinite set for which there exists some $\epsilon > 0$ such that in any subset of $A$ of size $n$ there is a subset of size at least $\epsilon n$ which contains no three-term arithmetic progression.disprovedFormalized	disproved	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
#857	Estimate `m(n,k)`, or better give an asymptotic formula.openFormalized	open	In Lean	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

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