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Erdős Problems

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

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  • Open per source608
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Problems

1,217 Problems · 2 with reviewed evidence

9/26

NumberQuestionOpen
#385Note that trivially F(n)≤n+nF(n) \leq n + \sqrt{n}.openFormalized
#386There is a kk, such that 2≤k≤n−22 \le k \le n - 2 and (nk)\binom{n}{k} can be the product of consecutive primes infinitely often?openFormalized
#387Is there an absolute constant c>0c > 0 such that, for all 1≤k<n1 \leq k < n, the binomial coefficient (nk)\binom{n}{k} has a divisor in (cn,n](cn, n]?solvedFormalized
#388No statement retained — open to read what the source holdsopenNo formal declaration
#389Is it true that for every n≥1n \geq 1 there is a kk such that n(n+1)⋯(n+k−1)∣(n+k)⋯(n+2k−1)? n(n + 1) \cdots (n + k - 1) \mid (n + k) \cdots (n + 2k - 1)? openFormalized
#390Does there exists a constant c such that f n - 2 * n ~ c * (n / log n)?openFormalized
#391No statement retained — open to read what the source holdsprovedNo formal declaration
#392Let A(n)A(n) denote the least value of tt such that n!=a1⋯at n! = a_1 \cdots a_t with a1≤⋯≤at≤n2a_1 \leq \cdots \leq a_t\leq n^2. Then A(n)=n2−n2log⁡n+o(nlog⁡n). A(n) = \frac{n}{2} - \frac{n}{2\log n} + o\left(\frac{n}{\log n}\right). proved (Lean)Formalized
#393No statement retained — open to read what the source holdsopenNo formal declaration
#394t k n = v when v works and nothing positive below it does.openFormalized
#395No statement retained — open to read what the source holdsprovedNo formal declaration
#396Is it true that for every kk there exists nn such that ∏0≤i≤k(n−i)∣(2nn)?\prod_{0\leq i\leq k}(n-i) \mid \binom{2n}{n}?openFormalized
#397Are there only finitely many solutions to ∏i(2mimi)=∏j(2njnj) \prod_i \binom{2m_i}{m_i}=\prod_j \binom{2n_j}{n_j} with the mi,njm_i,n_j distinct?disproved (Lean)Formalized
#398Brocard's Problem Does n!+1=m2n! + 1 = m^2 have integer solutions other than n=4,5,7n = 4, 5, 7?falsifiableFormalized
#399Is it true that there are no solutions to n! = x^k ± y^k with x,y,n ∈ ℕ, x*y > 1, and k > 2?disproved (Lean)Formalized
#400Can one show that ∑n≤xgk(n)∼ckxlog⁡x\sum_{n\leq x}g_k(n) \sim c_k x\log x for some constant ckc_k?openFormalized
#401Is there some function f(r)f(r) such that f(r)→∞f(r)\to \infty as r→∞r\to\infty, such that, for infinitely many nn, there exist a1,a2a_1,a_2 with a1+a2>n+f(r)log⁡na_1+a_2> n+f(r)\log n such that a1!a2!∣n!2n3n⋯prna_1!a_2! \mid n!2^n3^n\cdots p_r^n?proved (Lean)Formalized
#402Prove that, for any finite set A⊂NA\subset\mathbb{N}, there exist a,b∈Aa, b\in A such that gcd⁡(a,b)≤a/∣A∣. \gcd(a, b)\leq a/|A|. provedFormalized
#403Does the equation 2m=a1!+⋯+ak!2^m=a_1!+\cdots+a_k! with a1<a2<⋯<aka_1<a_2<\cdots <a_k have only finitely many solutions?proved (Lean)Formalized
#404No statement retained — open to read what the source holdsopenNo formal declaration
#405No statement retained — open to read what the source holdsprovedNo formal declaration
#406Is it true that there are only finitely many powers of 22 which have only the digits 00 and 11 when written in base 33?openFormalized
#407No statement retained — open to read what the source holdsprovedNo formal declaration
#408No statement retained — open to read what the source holdsopenNo formal declaration
#409How many iterations of n↦ϕ(n)+1n\mapsto\phi(n) + 1 are needed before a prime is reached?openFormalized
#410Let σ1(n)=σ(n)σ_1(n) = σ(n), the sum of divisors function, and σk(n)=σ(σk−1(n))σ_k(n) = σ(σ_{k-1}(n)).openFormalized
#411No statement retained — open to read what the source holdsopenNo formal declaration
#412Let σ1(n)=σ(n)σ_1(n)=σ(n), the sum of divisors function, and σk(n)=σ(σk−1(n))σ_k(n) = σ(σ_{k-1}(n)). Is it true that, for every m,n≥2m, n ≥ 2, there exist some i,ji, j such that σi(m)=σj(n)σ_i(m) = σ_j(n)?openFormalized
#413Are there infinitely many barriers for ω?openFormalized
#414Let h1(n)=h(n)h_1(n) = h(n) and hk(n)=h(hk−1(n))h_k(n) = h(h_{k-1}(n)). Is it true, for any m,nm,n, there exist ii and jj such that hi(m)=hj(n)h_i(m) = h_j(n)?openFormalized
#415No statement retained — open to read what the source holdsopenNo formal declaration
#416Let V(x) count the number of n≤x such that ϕ(m)=n is solvable. Does V(2x)/V(x)→2 ?openFormalized
#417LetV′(x)=#{ϕ(m):1≤m≤x}V'(x)=\#\{\phi(m) : 1\leq m\leq x\}andV(x)=#{ϕ(m)≤x:1≤m}.V(x)=\#\{\phi(m) \leq x : 1\leq m\}. Does lim⁡V(x)/V′(x)\lim V(x)/V'(x) exist?openFormalized
#418Are there infinitely many integers not of the form n−ϕ(n)n - \phi(n)?proved (Lean)Formalized
#419If τ(n)\tau(n) counts the number of divisors of nn, then what is the set of limit points of τ((n+1)!)τ(n!)? \frac{\tau((n+1)!)}{\tau(n!)}? solved (Lean)Formalized
#420No statement retained — open to read what the source holdsopenNo formal declaration
#421Is there a sequence 1≤d1<d2<…1 \le d_1 < d_2 < \dots with density 1 such that all products ∏u≤i≤vdi\prod_{u \le i \le v} d_i are distinct?openFormalized
#422Does f(n)f(n) miss infinitely many integers?openFormalized
#423No statement retained — open to read what the source holdsopenNo formal declaration
#424Let a1=2a_1 = 2 and a2=3a_2 = 3 and continue the sequence by appending to a1,…,ana_1, \ldots, a_n all possible values of aiaj−1a_i a_j - 1 with i≠ji \neq j. Is it true that the set of integers which eventually appear has positive density?openFormalized
#425No statement retained — open to read what the source holdsopenNo formal declaration
#426We say HH is a unique subgraph of GG if there is exactly one way to find HH as a subgraph (not necessarily induced) of GG. Is there a graph on nn vertices with ≫2(n2)n!\gg \frac{2^{\binom{n}{2}}}{n!} many distinct unique subgraphs?disproved (Lean)Formalized
#427Erdős Problem 427: is it true that, for every nn and dd, there exists kk such that d∣pn+1+⋯+pn+k, d \mid p_{n + 1} + \cdots + p_{n + k}, where prp_r denotes the rrth prime?proved (Lean)Formalized
#428Is there a set A⊆NA\subseteq \mathbb{N} such that, for infinitely many nn, all of n−an-a are prime for all a∈Aa\in A with 0<a<n0 < a < n and lim inf⁡∣A∩[1,x]∣π(x)>0?\liminf\frac{\lvert A\cap [1,x]\rvert}{\pi(x)}>0?openFormalized
#429Is it true that, if A⊆NA\subseteq \mathbb{N} is sparse enough and does not cover all residue classes modulo pp for any prime pp, then there exists some nn such that n+an+a is prime for all a∈Aa\in A?disproved (Lean)Formalized
#430No statement retained — open to read what the source holdsopenNo formal declaration
#431No statement retained — open to read what the source holdsopenNo formal declaration
#432No statement retained — open to read what the source holdsopenNo formal declaration

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