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

12/12

Number	Question	Source says	Formal	Result here
#1099	No statement retained — open to read what the source holdsprovedNo formal declaration	proved	No formal statement	No Result decision
#1100	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#1101	1. There is NO good sequence with polynomial growth.openFormalized	open	In Lean	No Result decision
#1102	If `A = {a₁ < a₂ < …}` has property P, then `A` has natural density `0`. Equivalently, `(a_j / j) → ∞` as `j → ∞`. -solved (Lean)Formalized	solved (Lean)	In Lean	No Result decision
#1103	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#1106	Let $p(n)$ be the partition number of $n$ and $F(n)$ be the number of distinct prime factors of $∏_{i= 1} ^ {n} p(n)$ , then $F(n)$ tends to infinity when $n$ tends to infinity.openFormalized	open	In Lean	No Result decision
#1107	Let $r \ge 2$ . Is every large integer the sum of at most $r + 1$ many $r$ -powerful numbers?openFormalized	open	In Lean	No Result decision
#1108	For each $k \geq 2$ , does the set $A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}\text{ finite}\right\}$ of all finite sums of distinct factorials contain only finitely many $k$ -th powers?openFormalized	open	In Lean	No Result decision
#1109	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#1110	Let $p>q\geq 2$ be two coprime integers. We call $n$ representable if it is the sum of integers of the form $p^kq^l$ , none of which divide each other.openFormalized	open	In Lean	No Result decision
#1113	Erdős Problem 1113. Do there exist Sierpiński numbers that possess no finite covering set of primes?openFormalized	open	In Lean	No Result decision
#1122	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#1134	No statement retained — open to read what the source holdsdisproved (Lean)No formal declaration	disproved (Lean)	No formal statement	No Result decision
#1135	The Collatz conjecture states that for any positive integer $n$ , there exists a natural number $m$ such that the $m$ -th term of the sequence is 1.openFormalized	open	In Lean	No Result decision
#1136	Does there exist $A\subset \mathbb{N}$ with lower density $>1/3$ such that $a+b\neq 2^k$ for any $a,b\in A$ and $k\geq 0$ ?proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#1137	Let $d_n=p_{n+1}-p_n$ , where $p_n$ denotes the $n$ th prime. Is it true that $\frac{\max_{n < x}d_{n}d_{n-1}}{(\max_{n < x}d_n)^2}\to 0$ as $x\to \infty$ ?openFormalized	open	In Lean	No Result decision
#1138	Erdős Problem 1138. Let $x/2 < y < x$ and $C > 1$ . If $d = \max_{p_n < x}(p_{n+1} - p_n)$ , where $p_n$ denotes the $n$ -th prime, then is it true that $\pi(y + Cd) - \pi(y) \sim \frac{Cd}{\log y}$ ?disproved (Lean)Formalized	disproved (Lean)	In Lean	No Result decision
#1139	Let $1\leq u_1 < u_2 < \cdots$ be the sequence of integers with at most $2$ prime factors. Is it true that $\limsup_{k \to \infty} \frac{u_{k+1}-u_k}{\log k}=\infty?$ openFormalized	open	In Lean	No Result decision
#1140	No statement retained — open to read what the source holdsdisprovedNo formal declaration	disproved	No formal statement	No Result decision
#1141	Are there infinitely many $n$ such that $n-k^2$ is prime for all $k$ with $(n,k)=1$ and $k^2 < n$ ?disproved (Lean)Formalized	disproved (Lean)	In Lean	No Result decision
#1142	Are there infinitely many $n > 2$ such that $n - 2^k$ is prime for all $k \geq 1$ with $2^k < n$ ?openFormalized	open	In Lean	No Result decision
#1143	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#1144	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#1146	Is $B=\{2^m3^n : m,n\geq 0\}$ an essential component?openFormalized	open	In Lean	No Result decision
#1148	Can every large integer $n$ be written as $n=x^2+y^2-z^2$ with $\max(x^2,y^2,z^2)\leq n$ ?proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#1149	No statement retained — open to read what the source holdsprovedNo formal declaration	proved	No formal statement	No Result decision
#1180	No statement retained — open to read what the source holdsprovedNo formal declaration	proved	No formal statement	No Result decision
#1181	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#1184	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#1187	No statement retained — open to read what the source holdssolvedNo formal declaration	solved	No formal statement	No Result decision
#1188	Call a set of distinct integers $1<n_1<\cdots<n_k$ with associated congruence classes $a_i\pmod{n_i}$ a distinct covering system if every integer satisfies at least one of these congruences. A minimal distinct covering system is one such that no proper subset forms a covering system. Let $F(x)$ count the number of minimal distinct covering systems with all moduli in $[1,x]$ . Estimate $F(x)$ .openFormalized	open	In Lean	No Result decision
#1189	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#1190	Let $\epsilon_m=\max \sum \frac{1}{n_i}$ where the maximum is taken over all finite sequences $m<n_1<\cdots<n_k$ for which there exist congruences $a_i\pmod{n_i}$ such that no integer satisfies two such congruences.solved (Lean)Formalized	solved (Lean)	In Lean	No Result decision
#1195	No statement retained — open to read what the source holdssolvedNo formal declaration	solved	No formal statement	No Result decision
#1196	Is it true that, for any $x$ , if $A\subset [x,\infty)$ is a primitive set of integers (so that no distinct elements of $A$ divide each other) then $\sum_{a\in A}\frac{1}{a\log a}< 1+o(1),$ where the $o(1)$ term $\to 0$ as $x\to \infty$ ? -proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#1200	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#1201	Is it true that for every $\epsilon,\eta>0$ there exists a $k$ such that the density of $n$ for which $P(n(n+1)\cdots(n+k))>n^{1-\epsilon}$ is at least $1-\eta$ (where $P(m)$ is the greatest prime divisor of $m$ )?openFormalized	open	In Lean	No Result decision
#1202	No statement retained — open to read what the source holdssolvedNo formal declaration	solved	No formal statement	No Result decision
#1203	Prove that $F(n)\to \infty$ as $n\to \infty$ .openFormalized	open	In Lean	No Result decision
#1204	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#1205	No statement retained — open to read what the source holdssolvedNo formal declaration	solved	No formal statement	No Result decision
#1206	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#1209	Let $A=\{a_1<a_2<\cdots\}$ be a sequence of integers which tends to infinity sufficiently fast. If there is an $n$ such that all $n+a_k$ are primes then must there exist infinitely many such $n$ ?openFormalized	open	In Lean	No Result decision
#1210	Let $A\subseteq [1,n)$ be a set of integers such that $(a,b)=1$ for all distinct $a,b\in A$ . Is it true that $\sum_{a\in A}\frac{1}{n-a}\leq \sum_{p < n}\frac{1}{p}+O(1)$ ?openFormalized	open	In Lean	No Result decision
#1211	No statement retained — open to read what the source holdssolvedNo formal declaration	solved	No formal statement	No Result decision
#1212	Roughness criterion (sufficiency for the anchor conditions): if $a < s$ for all $s$ in the leg and the leg stays below $a + P^-(a)$ , then $a$ is coprime to the whole leg. Stated via divisibility: no prime factor of $a$ divides any $s$ with $a < s < a + p$ for all prime factors $p$ of $a$ .openFormalized	open	In Lean	No Result decision
#1214	Let $x,y\geq 1$ be integers such that, for all $n\geq 1$ , the set of primes dividing $x^{n}-1$ is equal to the set of primes dividing $y^n-1$ . Must $x=y$ ?provedFormalized	proved	In Lean	No Result decision
#1217	No statement retained — open to read what the source holdsprovedNo formal declaration	proved	No formal statement	No Result decision

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