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

1,217 Problems · 2 with reviewed evidence

9/26

Number	Question	Source says	Formal	Result here
#385	Note that trivially $F(n) \leq n + \sqrt{n}$ .Source says openFormalized	open	In Lean	No Result decision
#386	There is a $k$ , such that $2 \le k \le n - 2$ and $\binom{n}{k}$ can be the product of consecutive primes infinitely often?Source says openFormalized	open	In Lean	No Result decision
#387	Is there an absolute constant $c > 0$ such that, for all $1 \leq k < n$ , the binomial coefficient $\binom{n}{k}$ has a divisor in $(cn, n]$ ?Source says solvedFormalized	solved	In Lean	No Result decision
#388	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#389	Is it true that for every $n \geq 1$ there is a $k$ such that $n(n + 1) \cdots (n + k - 1) \mid (n + k) \cdots (n + 2k - 1)?$ Source says openFormalized	open	In Lean	No Result decision
#390	Does there exists a constant `c` such that `f n - 2 * n ~ c * (n / log n)`?Source says openFormalized	open	In Lean	No Result decision
#391	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision
#392	Let $A(n)$ denote the least value of $t$ such that $n! = a_1 \cdots a_t$ with $a_1 \leq \cdots \leq a_t\leq n^2$ . Then $A(n) = \frac{n}{2} - \frac{n}{2\log n} + o\left(\frac{n}{\log n}\right).$ Source says proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#393	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#394	`t k n = v` when `v` works and nothing positive below it does.Source says openFormalized	open	In Lean	No Result decision
#395	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision
#396	Is it true that for every $k$ there exists $n$ such that $\prod_{0\leq i\leq k}(n-i) \mid \binom{2n}{n}?$ Source says openFormalized	open	In Lean	No Result decision
#397	Are there only finitely many solutions to $\prod_i \binom{2m_i}{m_i}=\prod_j \binom{2n_j}{n_j}$ with the $m_i,n_j$ distinct?Source says disproved (Lean)Formalized	disproved (Lean)	In Lean	No Result decision
#398	Brocard's Problem Does $n! + 1 = m^2$ have integer solutions other than $n = 4, 5, 7$ ?Source says falsifiableFormalized	falsifiable	In Lean	No Result decision
#399	Is it true that there are no solutions to `n! = x^k ± y^k` with `x,y,n ∈ ℕ`, `x*y > 1`, and `k > 2`?Source says disproved (Lean)Formalized	disproved (Lean)	In Lean	No Result decision
#400	Can one show that $\sum_{n\leq x}g_k(n) \sim c_k x\log x$ for some constant $c_k$ ?Source says openFormalized	open	In Lean	No Result decision
#401	Is there some function $f(r)$ such that $f(r)\to \infty$ as $r\to\infty$ , such that, for infinitely many $n$ , there exist $a_1,a_2$ with $a_1+a_2> n+f(r)\log n$ such that $a_1!a_2! \mid n!2^n3^n\cdots p_r^n$ ?Source says proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#402	Prove that, for any finite set $A\subset\mathbb{N}$ , there exist $a, b\in A$ such that $\gcd(a, b)\leq a/\|A\|.$ Source says provedFormalized	proved	In Lean	No Result decision
#403	Does the equation $2^m=a_1!+\cdots+a_k!$ with $a_1<a_2<\cdots <a_k$ have only finitely many solutions?Source says proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#404	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#405	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision
#406	Is it true that there are only finitely many powers of $2$ which have only the digits $0$ and $1$ when written in base $3$ ?Source says openFormalized	open	In Lean	No Result decision
#407	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision
#408	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#409	How many iterations of $n\mapsto\phi(n) + 1$ are needed before a prime is reached?Source says openFormalized	open	In Lean	No Result decision
#410	Let $σ_1(n) = σ(n)$ , the sum of divisors function, and $σ_k(n) = σ(σ_{k-1}(n))$ .Source says openFormalized	open	In Lean	No Result decision
#411	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#412	Let $σ_1(n)=σ(n)$ , the sum of divisors function, and $σ_k(n) = σ(σ_{k-1}(n))$ . Is it true that, for every $m, n ≥ 2$ , there exist some $i, j$ such that $σ_i(m) = σ_j(n)$ ?Source says openFormalized	open	In Lean	No Result decision
#413	Are there infinitely many barriers for `ω`?Source says openFormalized	open	In Lean	No Result decision
#414	Let $h_1(n) = h(n)$ and $h_k(n) = h(h_{k-1}(n))$ . Is it true, for any $m,n$ , there exist $i$ and $j$ such that $h_i(m) = h_j(n)$ ?Source says openFormalized	open	In Lean	No Result decision
#415	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#416	Let `V(x)` count the number of `n≤x` such that `ϕ(m)=n` is solvable. Does `V(2x)/V(x)→2` ?Source says openFormalized	open	In Lean	No Result decision
#417	Let $V'(x)=\#\{\phi(m) : 1\leq m\leq x\}$ and $V(x)=\#\{\phi(m) \leq x : 1\leq m\}.$ Does $\lim V(x)/V'(x)$ exist?Source says openFormalized	open	In Lean	No Result decision
#418	Are there infinitely many integers not of the form $n - \phi(n)$ ?Source says proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#419	If $\tau(n)$ counts the number of divisors of $n$ , then what is the set of limit points of $\frac{\tau((n+1)!)}{\tau(n!)}?$ Source says solved (Lean)Formalized	solved (Lean)	In Lean	No Result decision
#420	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#421	Is there a sequence $1 \le d_1 < d_2 < \dots$ with density 1 such that all products $\prod_{u \le i \le v} d_i$ are distinct?Source says openFormalized	open	In Lean	No Result decision
#422	Does $f(n)$ miss infinitely many integers?Source says openFormalized	open	In Lean	No Result decision
#423	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#424	Let $a_1 = 2$ and $a_2 = 3$ and continue the sequence by appending to $a_1, \ldots, a_n$ all possible values of $a_i a_j - 1$ with $i \neq j$ . Is it true that the set of integers which eventually appear has positive density?Source says openFormalized	open	In Lean	No Result decision
#425	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#426	We say $H$ is a unique subgraph of $G$ if there is exactly one way to find $H$ as a subgraph (not necessarily induced) of $G$ . Is there a graph on $n$ vertices with $\gg \frac{2^{\binom{n}{2}}}{n!}$ many distinct unique subgraphs?Source says disproved (Lean)Formalized	disproved (Lean)	In Lean	No Result decision
#427	Erdős Problem 427: is it true that, for every $n$ and $d$ , there exists $k$ such that $d \mid p_{n + 1} + \cdots + p_{n + k},$ where $p_r$ denotes the $r$ th prime?Source says proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#428	Is there a set $A\subseteq \mathbb{N}$ such that, for infinitely many $n$ , all of $n-a$ are prime for all $a\in A$ with $0 < a < n$ and $\liminf\frac{\lvert A\cap [1,x]\rvert}{\pi(x)}>0?$ Source says openFormalized	open	In Lean	No Result decision
#429	Is it true that, if $A\subseteq \mathbb{N}$ is sparse enough and does not cover all residue classes modulo $p$ for any prime $p$ , then there exists some $n$ such that $n+a$ is prime for all $a\in A$ ?Source says disproved (Lean)Formalized	disproved (Lean)	In Lean	No Result decision
#430	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#431	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#432	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision