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

11/12

Number	Question	Source says	Formal	Result here
#962	Main conjecture:Source says openFormalized	open	In Lean	No Result decision
#963	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#964	No statement retained — open to read what the source holdsSource says proved (Lean)No formal declaration	proved (Lean)	No formal statement	No Result decision
#966	Let $k,r\geq 2$ . Does there exist a set $A\subseteq \mathbb{N}$ that contains no non-trivial arithmetic progression of length $k+1$ , yet in any $r$ -colouring of $A$ there must exist a monochromatic non-trivial arithmetic progression of length $k$ ?Source says proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#967	Let $1<a_1<\cdots$ be a sequence of integers such that $\sum\frac{1}{a_i}<\infty$ . Is it true that, for every $t\in \mathbb{R}$ , $1+\sum_{k}\frac{1}{a_k^{1+it}}\neq 0?$ Source says disproved (Lean)Formalized	disproved (Lean)	In Lean	No Result decision
#968	Does the set `{n \| u n < u (n+1)}` have positive lower density?Source says openFormalized	open	In Lean	No Result decision
#969	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#970	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#971	Let `p(a, d)` be the least prime congruent to `a (mod d)`. Does there exist a constant `c > 0` such that for all large `d`, `p(a, d) > (1 + c) * φ(d) * log d` for `≫ φ(d)` many values of `a`?Source says openFormalized	open	In Lean	No Result decision
#972	Erdős problem 972. Let $\alpha > 1$ be irrational. Are there infinitely many primes $p$ such that $\lfloor p\alpha \rfloor$ is also prime?Source says openFormalized	open	In Lean	No Result decision
#975	For an irreducible polynomial $f \in \mathbb{Z}[x]$ with $f(n) \ge 1$ for sufficiently large $n$ , does there exists a constant $c = c(f) > 0$ such that $\sum_{n \le x} \tau(f(n)) \approx c \cdot x \log x$ ?Source says openFormalized	open	In Lean	No Result decision
#976	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#977	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision
#978	Let `f ∈ ℤ[X]` be an irreducible polynomial with positive leading coefficient. Suppose that the degree `k` of `f` is larger than `2`, is not equal to a power of `2`, and `f n` has no fixed `(k - 1)`-th power divisors other than `1`. Then the set of `n` such that `f n` is `(k - 1)`-th power free has positive density, and this is proved in [Ho67].Source says openFormalized	open	In Lean	No Result decision
#979	Let $k ≥ 2$ , and let $f_k(n)$ count the number of solutions to $n = p_1^k + \dots + p_k^k$ , where the $p_i$ are prime numbers. Is it true that $\limsup f_k(n) = \infty$ ?Source says openFormalized	open	In Lean	No Result decision
#980	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision
#981	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision
#983	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#985	Is it true that, for every prime $p$ , there is a prime $q \leq p$ which is a primitive root modulo $p$ ?Source says openFormalized	open	In Lean	No Result decision
#999	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision
#1000	Let $A=\{n_1<n_2<\cdots\}$ be an infinite sequence of integers, and let $\phi_A(k)$ count the number of $1\leq m\leq n_k$ such that the fraction $\frac{m}{n_k}$ does not have denominator $n_j$ for $j<k$ when written in lowest form; equivalently, $\frac{n_k}{(m,n_k)}\neq n_j$ for all $1\leq j<k$ .Source says proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#1001	No statement retained — open to read what the source holdsSource says solvedNo formal declaration	solved	No formal statement	No Result decision
#1003	Are there infinitely many solutions to $\phi(n) = \phi(n+1)$ , where $\phi$ is the Euler totient function?Source says openFormalized	open	In Lean	No Result decision
#1004	For any fixed c > 0, if x is sufficiently large then there exists n ≤ x such that the values of φ(n+k) are all distinct for 1 ≤ k ≤ (log x)^c. This is an open problem.Source says openFormalized	open	In Lean	No Result decision
#1005	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#1052	Are there only finitely many unitary perfect numbers?Source says openFormalized	open	In Lean	No Result decision
#1053	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#1054	Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$ . Show that $f$ is undefined at $n=2$ , i.e. we get the junk value $0$ .Source says openFormalized	open	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
#1056	Let $k ≥ 2$ . Does there exist a prime $p$ and consecutive intervals $I_0,\dots,I_k$ such that $\prod\limits_{n{\in}I_i}n \equiv 1 \mod n$ for all $1 \le i \le k$ ?Source says openFormalized	open	In Lean	No Result decision
#1057	Is it true that $C(x)=x^{1-o(1)}$ ?Source says openFormalized	open	In Lean	No Result decision
#1058	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision
#1059	Are there infinitely many primes $p$ such that $p - k!$ is composite for each $k$ such that $1 ≤ k! < p$ ?Source says openFormalized	open	In Lean	No Result decision
#1060	The conjecture is about the function $f(n)$ which counts the number of solutions to $k\sigma(k)=n$ , where $\sigma(k)$ is the sum of divisors of $k$ . The first bound is that $f(n)$ grows slower than any power of $n^(\frac{1}{\log\log n})$ . The second bound is that $f(n)$ is at most a power of $\log n$ .Source says openFormalized	open	In Lean	No Result decision
#1061	How many (ordered) solutions are there to `σ(a) + σ(b) = σ(a + b)` with `a + b ≤ x`? Is it true that this number is asymptotic to `c * x` for some constant `c > 0`?Source says openFormalized	open	In Lean	No Result decision
#1062	Erdős asked whether the limiting density `f n / n` exists and, if so, whether it is irrational.Source says openFormalized	open	In Lean	No Result decision
#1063	Estimate $n_k$ by finding a better upper bound.Source says openFormalized	open	In Lean	No Result decision
#1064	Let $ϕ(n)$ be the Euler's totient function, then the $n$ satisfies $ϕ(n)>ϕ(n - ϕ(n))$ have asymptotic density 1. Reference: [LuPo02] Luca, Florian and Pomerance, Carl, On some problems of {M}\polhk akowski-{S}chinzel and {E}rdős concerning the arithmetical functions { $\phi$ } and { $\sigma$ }. Colloq. Math.Source says provedFormalized	proved	In Lean	No Result decision
#1065	Are there infinitely many primes $p$ such that $p = 2^k * q + 1$ for some prime $q$ and $k ≥ 0$ ?Source says openFormalized	open	In Lean	No Result decision
#1072	Is it true that there are infinitely many $p$ for which $f(p) = p − 1$ ?Source says openFormalized	open	In Lean	No Result decision
#1073	Is it true that $A(x) \le x^{o(1)}$ ?Source says openFormalized	open	In Lean	No Result decision
#1074	Let $S$ be the set of all $m\geq 1$ such that there exists a prime $p\not\equiv 1\pmod{m}$ such that $m! + 1 \equiv 0\pmod{p}$ . Does $\lim\frac{\|S\cap[1, x]\|}{x}$ exist?Source says openFormalized	open	In Lean	No Result decision
#1081	No statement retained — open to read what the source holdsSource says disprovedNo formal declaration	disproved	No formal statement	No Result decision
#1093	Are there infinitely many binomial coefficients with deficiency 1?Source says openFormalized	open	In Lean	No Result decision
#1094	For all $n\ge 2k$ the least prime factor of $\binom{n}{k}$ is $\le\max(n/k,k)$ , with only finitely many exceptions.Source says openFormalized	open	In Lean	No Result decision
#1095	Sorenson, Sorenson, and Webster [SSWE20] give heuristic evidence that $\log g(k) \asymp \frac{k}{\log k}$ .Source says openFormalized	open	In Lean	No Result decision
#1096	Let $1<q<1+\epsilon$ and consider the set of numbers of the shape $\sum_{i\in S}q^i$ (for all finite $S$ ), ordered by size as $0=x_1<x_2<\cdots$ .Source says provedFormalized	proved	In Lean	No Result decision
#1097	The main conjecture: for any finite set of integers $A$ with $\|A\| = n$ , the number of distinct common differences in three-term arithmetic progressions is $O(n^{3/2})$ .Source says openFormalized	open	In Lean	No Result decision