Erdős Problems

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

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22 Problems

Number	Question	Source says	Formal	Result here
#2	No statement retained — open to read what the source holdsSource says disprovedNo formal declaration	disproved	No formal statement	No Result decision
#7	Is there a covering system all of whose moduli are odd (and greater than 1)?Source says verifiableFormalized	verifiable	In Lean	No Result decision
#8	No statement retained — open to read what the source holdsSource says disprovedNo formal declaration	disproved	No formal statement	No Result decision
#27	No statement retained — open to read what the source holdsSource says disprovedNo formal declaration	disproved	No formal statement	No Result decision
#202	Let $n_1<\cdots < n_r\leq N$ with associated $a_i\pmod{n_i}$ such that the congruence classes are disjoint (that is, every integer is $\equiv a_i\pmod{n_i}$ for at most one $1\leq i\leq r$ ). How large can $r$ be in terms of $N$ ?Source says solved (Lean)Formalized	solved (Lean)	In Lean	No Result decision
#203	Is there an integer $m$ with $(m, 6) = 1$ such that none of $2^k \cdot 3^\ell \cdot m + 1$ are prime, for any $k, \ell \ge 0$ ?Source says openFormalized	open	In Lean	No Result decision
#204	Are there $n$ such that there is a covering system with moduli the divisors of $n$ which is 'as disjoint as possible'?Source says disproved (Lean)Formalized	disproved (Lean)	In Lean	No Result decision
#273	Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p \geq 5$ ?Source says openFormalized	open	In Lean	No Result decision
#274	If $G$ is a group, can there exist an exact covering of $G$ by more than one coset of different sizes? (i.e. each element is contained in exactly one of the cosets.)Source says openFormalized	open	In Lean	No Result decision
#275	If a finite system of $r$ congruences $\{ a_i\pmod{n_i} : 1\leq i\leq r\}$ (the $n_i$ are not necessarily distinct) covers $2^r$ consecutive integers then it covers all integers.Source says proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#276	Is there an infinite Lucas sequence $a_0, a_1, \ldots$ where $a_{n+2} = a_{n+1} + a_n$ for $n \ge 0$ such that all $a_k$ are composite, and yet no integer has a common factor with every term of the sequence?Source says openFormalized	open	In Lean	No Result decision
#277	Is it true that, for every $c$ , there exists an $n$ such that $\sigma(n)>cn$ but there is no covering system whose moduli all divide $n$ ?Source says provedFormalized	proved	In Lean	No Result decision
#278	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#279	Let $k\geq 3$ . Is there a choice of congruence classes $a_p\pmod{p}$ for every prime $p$ such that all sufficiently large integers can be written as $a_p+tp$ for some prime $p$ and integer $t\geq k$ ?Source says openFormalized	open	In Lean	No Result decision
#280	Let $n_1<n_2<\cdots$ be an infinite sequence of integers with associated $a_k\pmod{n_k}$ , such that for some $\epsilon>0$ we have $n_k>(1+\epsilon)k\log k$ for all $k$ . Then $\#\{ m<n_k : m\not\equiv a_i\pmod{n_i} \textrm{ for }1\leq i\leq k\}\neq o(k).$ Source says disproved (Lean)Formalized	disproved (Lean)	In Lean	No Result decision
#281	Let $n_1<n_2<\cdots$ be an infinite sequence such that, for any choice of congruence classes $a_i\pmod{n_i}$ , the set of integers not satisfying any of the congruences $a_i\pmod{n_i}$ has density $0$ . Is it true that for every $\epsilon>0$ there exists some $k$ such that, for every choice of congruence classes $a_i$ , the density of integers not satisfying any of the congruences $a_i\pmod{n_i}$ for $1\leq i\leq k$ is less than $\epsilon$ ?Source says proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#586	No statement retained — open to read what the source holdsSource says disprovedNo formal declaration	disproved	No formal statement	No Result decision
#947	No statement retained — open to read what the source holdsSource says proved (Lean)No formal declaration	proved (Lean)	No formal statement	No Result decision
#1113	Erdős Problem 1113. Do there exist Sierpiński numbers that possess no finite covering set of primes?Source says openFormalized	open	In Lean	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 $F(x)$ .Source says openFormalized	open	In Lean	No Result decision
#1189	No statement retained — open to read what the source holdsSource says openNo 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.Source says solved (Lean)Formalized	solved (Lean)	In Lean	No Result decision