Erdős Problems

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

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61 Problems

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Number	Question	Source says	Formal	Result here
#4	Is it true that, for any $C > 0$ , there infinitely many $n$ such that: $p_{n + 1} - p_n > C \frac{\log\log n\log\log\log\log n}{(\log\log\log n) ^ 2}\log n$ Source says provedFormalized	proved	In Lean	No Result decision
#5	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#6	There are infinitely many $n$ such that $d_n < d_{n+1} < d_{n+2}$ , where $d$ denotes the prime gap function.Source says provedFormalized	proved	In Lean	No Result decision
#9	Is the upper density of the set of odd numbers that cannot be expressed as a prime plus two powers of 2 positive?Source says openFormalized	open	In Lean	No Result decision
#10	Is there some $k$ such that every integer is the sum of a prime and at most $k$ powers of $2$ ?Source says openFormalized	open	In Lean	No Result decision
#15	Is it true that $\sum_{n=1}^\infty(-1)^n\frac{n}{p_n}$ converges, where $p_n$ is the sequence of primes?Source says openFormalized	open	In Lean	No Result decision
#16	Is the set of odd integers not of the form $2^k+p$ the union of an infinite arithmetic progression and a set of density $0$ ?Source says disproved (Lean)Formalized	disproved (Lean)	In Lean	No Result decision
#17	Erdős Problem 17. Are there infinitely many cluster primes?Source says openFormalized	open	In Lean	No Result decision
#49	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision
#141	Let $k≥3$ . Are there $k$ consecutive primes in arithmetic progression?Source says openFormalized	open	In Lean	No Result decision
#200	Does the longest arithmetic progression of primes in $\{1,\ldots,N\}$ have length $o(\log N)$ ?Source says openFormalized	open	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
#218	The set of indices $n$ for which a prime gap is preceded by a larger or equal prime gap has a natural density of $\frac 1 2$ .Source says openFormalized	open	In Lean	No Result decision
#219	Are there arbitrarily long arithmetic progressions of primes? Solution: yes. Ref: Green, Ben and Tao, Terence, _The primes contain arbitrarily long arithmetic progressions_Source says provedFormalized	proved	In Lean	No Result decision
#233	A conjecture by Heath-Brown: The sum of squares of the first $N$ gaps between consecutive primes behaves like $N * (log N)^2$ .Source says openFormalized	open	In Lean	No Result decision
#234	Is it true that for all `c ≥ 0`, the density `f c` of integers for which `(p (n + 1) - p n) / log n < c` exists and is a continuous function of `c`?Source says openFormalized	open	In Lean	No Result decision
#236	Let $f(n)$ count the number of solutions to $n=p+2^k$ for prime $p$ and $k\geq 0$ . Show that $f(n)=o(\log n)$ .Source says openFormalized	open	In Lean	No Result decision
#237	No statement retained — open to read what the source holdsSource says proved (Lean)No formal declaration	proved (Lean)	No formal statement	No Result decision
#238	Let `c₁, c₂ > 0`. Is it true that for any sufficiently large `x`, there exists more than `c₁ * log x` many consecutive primes `≤ x` such that the difference between any two is `> c₂`?Source says openFormalized	open	In Lean	No Result decision
#240	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision
#244	Let $C > 1$ . Does the set of integers of the form $p + \lfloor C^k \rfloor$ , for some prime $p$ and $k\geq 0$ , have density $>0$ ?Source says openFormalized	open	In Lean	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
#358	When $A_n = n$ , the function $f$ defined above counts the number of odd divisors of $n$ .Source says provedFormalized	proved	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
#431	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#454	Is it true that `limsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop = ⊤`?Source says openFormalized	open	In Lean	No Result decision
#458	Let $\operatorname{lcm}(1, \dots, n)$ denote the least common multiple of $\{1, \dots, n\}$ . Let $p_k$ be the $k$ -th prime. Is it true that for all $k \geq 1$ , $\operatorname{lcm}(1, \dots, p_{k+1}-1) < p_k \cdot \operatorname{lcm}(1, \dots, p_k)$ ?Source says falsifiableFormalized	falsifiable	In Lean	No Result decision
#459	Let $f(u)$ be the largest $v$ such that no $m\in (u,v)$ is composed entirely of primes dividing $uv$ . Estimate $f(u)$ .Source says solved (Lean)Formalized	solved (Lean)	In Lean	No Result decision
#461	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#462	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#463	Is there a function $f$ with $f(n)\to\infty$ as $n\to\infty$ such that, for all large $n$ , there is a composite number $m$ such that $n + f(n) < m < n + p(m)$ Here $p(m)$ is the least prime factor of $m$ .Source says openFormalized	open	In Lean	No Result decision
#680	Is it true that, for all sufficiently large $n$ , there exists some $k$ such that $p(n+k)>k^2+1,$ where $p(m)$ denotes the least prime factor of $m$ ?Source says openFormalized	open	In Lean	No Result decision
#681	Erdős problem 681. Is it true that for all large $n$ there exists $k$ such that $n + k$ is composite and $p(n+k) > k^2$ , where $p(m)$ is the least prime factor of $m$ ?Source says openFormalized	open	In Lean	No Result decision
#682	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision
#683	There exists $c > 0$ such that $P(n, k) > \min\{n-k+1, k^{1 + c}\}$ for all $0 < k < n$ .}Source says openFormalized	open	In Lean	No Result decision
#684	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#685	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#779	A Conjecture of Marian Deaconescu, see p.120 in https://doi.org/10.2307/2975810Source says falsifiableFormalized	falsifiable	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?Source says openFormalized	open	In Lean	No Result decision
#852	No statement retained — open to read what the source holdsSource says openNo 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 $n \le x$ .Source says openFormalized	open	In Lean	No Result decision
#855	Erdős Problem 855 (Segal's conjecture): $\pi(x + y) \le \pi(x) + \pi(y)$ for sufficiently large $x, y$ .Source says openFormalized	open	In Lean	No Result decision
#860	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#890	If $\omega_k(n)$ counts the number of distinct prime factors of $n$ which are $>k$ , then is it true that, for every $k\geq 1$ , $\liminf_{n\to \infty}\sum_{0\leq i < k}\omega_k(n+i)\leq k?$ Source says openFormalized	open	In Lean	No Result decision
#950	Is it true that $\liminf f(n)=1$ ?Source says openFormalized	open	In Lean	No Result decision
#997	Is it true that, for every $\alpha$ , the sequence $\{ \alpha p_n\}$ is not well-distributed, if $p_n$ is the sequence of primes?Source says proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#1055	A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$ . In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$ , with equality for at least one prime factor. Are there infinitely many primes in each class?Source says openFormalized	open	In Lean	No Result decision