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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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1,217 Problems · 2 with reviewed evidence

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#865There exists a constant C>0C>0 such that, for all large NN, if A⊆{1,…,N}A\subseteq \{1,\ldots,N\} has size at least 58N+C\frac{5}{8}N+C then there are distinct a,b,c∈Aa,b,c\in A such that a+b,a+c,b+c∈Aa+b,a+c,b+c\in A.Source says proved (Lean)Formalized
#866No statement retained — open to read what the source holdsSource says openNo formal declaration
#867Is it true that if A={a1<⋯<at}⊆{1,…,N}A=\{a_1<\cdots <a_t\}\subseteq \{1,\ldots,N\} has no solutions to ai+ai+1+⋯+aj∈Aa_i+a_{i+1}+\cdots+a_j\in A then ∣A∣≤N2+O(1)?\lvert A\rvert \leq \frac{N}{2}+O(1)?Source says disproved (Lean)Formalized
#868Let AA be an additive basis of order 22, let f(n)f(n) denote the number of ways in which nn can be written as the sum of two elements from AA. If f(n)→∞f(n) \to \infty as n→∞n \to \infty, then must AA contain a minimal additive basis of order 22?Source says solvedFormalized
#869No statement retained — open to read what the source holdsSource says disprovedNo formal declaration
#870No statement retained — open to read what the source holdsSource says openNo formal declaration
#871Let AA be an additive basis of order 22, and suppose 1A∗1A(n)→∞1_A\ast 1_A(n)\to \infty as n→∞n\to \infty. Can AA be partitioned into two disjoint additive bases of order 22?Source says disproved (Lean)Formalized
#872Each move claims exactly one pool element, so the minimax value never exceeds the number of already claimed elements plus the number of still unclaimed elements.Source says openFormalized
#873Let A={a1<a2<… }⊆NA = \{a_1 < a_2 < \dots\} \subseteq \mathbb{N} and let F(A,X,k)F(A,X,k) count the number of ii such that [ai,ai+1,…,ai+k−1]<X[a_i,a_{i+1}, \dots ,a_{i+k−1}] < X, where the left-hand side is the least common multiple. Is it true that, for every ϵ>0\epsilon > 0, there exists some kk such that F(A,X,k)<XϵF(A,X,k) < X^\epsilon?Source says openFormalized
#874No statement retained — open to read what the source holdsSource says provedNo formal declaration
#875No statement retained — open to read what the source holdsSource says openNo formal declaration
#876No statement retained — open to read what the source holdsSource says openNo formal declaration
#877No statement retained — open to read what the source holdsSource says provedNo formal declaration
#878No statement retained — open to read what the source holdsSource says openNo formal declaration
#879No statement retained — open to read what the source holdsSource says openNo formal declaration
#880No statement retained — open to read what the source holdsSource says provedNo formal declaration
#881Let A ⊂ ℕ be an additive basis of order k which is minimal in the sense that if B ⊂ A is any infinite set, then A \ B is not a basis of order k.Source says openFormalized
#882No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#883No statement retained — open to read what the source holdsSource says openNo formal declaration
#884For a natural number n, let 1=d1<…<dτ(n)=n1 = d_1 < \dotsc < d_{\tau(n)} = n denote the divisors of n in increasing order. Does it hold that ∑1≤i<j≤τ(n)1dj−di≪1+∑1≤i<τ(n)1di+1−di\sum_{1 \le i < j \le \tau(n)} \frac{1}{d_j - d_i} \ll 1 + \sum_{1 \le i < \tau(n)} \frac{1}{d_{i + 1} - d_i} for n→∞‘n \to \infty`, i.e. Unparsed notation: \sum_{1 \le i < j \le \tau(n)} \frac{1}{d_j - d_i} \in O \left( 1 + \sum_{1 \le i < \tau(n)} \frac{1}{d_{i + 1} - d_i}) Unparsed notation: \right)?Source says disproved (Lean)Formalized
#885Is it true that, for every k≥1k \geq 1, there exist integers N1<⋯<NkN_1 < \dots < N_k such that ∣∩iD(Ni)∣≥k|\cap_i D(N_i)| \geq k?Source says openFormalized
#886Let ϵ>0\epsilon>0. Is it true that, for all large nn, the number of divisors of nn in (n1/2,n1/2+n1/2−ϵ)(n^{1/2},n^{1/2}+n^{1/2-\epsilon}) is Oϵ(1)O_\epsilon(1)?Source says openFormalized
#887Is there an absolute constant KK such that, for every C>0C > 0, if nn is sufficiently large then nn has at most KK divisors in (n12,n12+Cn14)(n^{\frac{1}{2}}, n^{\frac{1}{2}} + C n^{\frac{1}{4}}).Source says openFormalized
#888What is the size of the largest A⊆{1,…,n}A\subseteq \{1,\ldots,n\} such that if a≤b≤c≤d∈Aa\leq b\leq c\leq d\in A are such that abcdabcd is a square then ad=bcad=bc?Source says solvedFormalized
#889Let v(n,k)v(n,k) count the prime factors of n+kn+k which do not divide n+in+i for 0≤i<k0\leq i < k. Is it true that v0(n)=max⁡k≥0v(n,k)→∞v_0(n)=\max_{k\geq 0}v(n,k)\to \infty as n→∞n\to \infty?Source says openFormalized
#890If ωk(n)\omega_k(n) counts the number of distinct prime factors of nn which are >k>k, then is it true that, for every k≥1k\geq 1, lim inf⁡n→∞∑0≤i<kωk(n+i)≤k?\liminf_{n\to \infty}\sum_{0\leq i < k}\omega_k(n+i)\leq k?Source says openFormalized
#891Let 2=p1<p2<⋯2=p_1 < p_2 < \cdots be the primes and k≥2k\geq 2. Is it true that, for all sufficiently large nn, there must exist an integer in [n,n+p1⋯pk)[n,n+p_1\cdots p_k) with >k>k many prime factors?Source says openFormalized
#892No statement retained — open to read what the source holdsSource says openNo formal declaration
#893Does the limit lim⁡n→∞f(2n)f(n)\lim_{n\to\infty} \frac{f(2n)}{f(n)} tend to infinity?Source says openFormalized
#894No statement retained — open to read what the source holdsSource says provedNo formal declaration
#895No statement retained — open to read what the source holdsSource says provedNo formal declaration
#896No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#897Let f(n)f(n) be an additive function (so that f(ab)=f(a)+f(b)f(ab)=f(a)+f(b) if (a,b)=1(a,b)=1 such that lim sup⁡p,kf(pk)/log⁡(pk)=∞\limsup_{p,k} f(p^k) / \log(p^k) = ∞. Is it true that lim sup⁡n(f(n+1)−f(n))/log⁡n=∞\limsup_n (f(n+1)−f(n))/ \log n = ∞?Source says disproved (Lean)Formalized
#898If A,B,C∈R2A,B,C\in \mathbb{R}^2 form a triangle and PP is a point in the interior then, if NN is where the perpendicular from PP to ABAB meets the triangle, and similarly for MM and LL, PA‾+PB‾+PC‾≥2(PM‾+PN‾+PL‾). \overline{PA}+\overline{PB}+\overline{PC}\geq 2(\overline{PM}+\overline{PN}+\overline{PL}). Source says proved (Lean)Formalized
#899Let A⊆NA\subseteq\mathbb{N} be an infinite set such that ∣A∩{1,...,N}∣=o(N)|A\cap \{1, ..., N\}| = o(N). Is it true that lim sup⁡N→∞∣(A−A)∩{1,...,N}∣∣A∩{1,...,N}∣=∞? \limsup_{N\to\infty}\frac{|(A - A)\cap \{1, ..., N\}|}{|A \cap \{1, ..., N\}|} = \infty? Source says provedFormalized
#900No statement retained — open to read what the source holdsSource says provedNo formal declaration
#901No statement retained — open to read what the source holdsSource says openNo formal declaration
#902No statement retained — open to read what the source holdsSource says openNo formal declaration
#903No statement retained — open to read what the source holdsSource says provedNo formal declaration
#904Let r≥2r\geq 2 and let tr(n)t_r(n) be the Turán number (the maximal number of edges in a graph on nn vertices with no Kr+1K_{r+1}).Source says proved (Lean)Formalized
#905Every graph with nn vertices and >n2/4>n^2/4 edges contains an edge which is in at least n/6n/6 triangles.Source says proved (Lean)Formalized
#906Does there exists an entire non-zero transcendental function f : ℂ → ℂ such that for any sequence n₀ < n₁ < ..., { z | ∃ k, iteratedDeriv (n k) f z = 0 } is dense.Source says openFormalized
#907Let f:R→Rf:\mathbb{R}\to \mathbb{R} be such that f(x+h)−f(x)f(x+h)-f(x) is continuous for every h>0h>0. Is it true that f=g+hf=g+h for some continuous gg and additive hh (i.e. h(x+y)=h(x)+h(y)h(x+y)=h(x)+h(y))?Source says proved (Lean)Formalized
#908No statement retained — open to read what the source holdsSource says provedNo formal declaration
#909No statement retained — open to read what the source holdsSource says provedNo formal declaration
#910No statement retained — open to read what the source holdsSource says disprovedNo formal declaration
#911No statement retained — open to read what the source holdsSource says openNo formal declaration
#912Prove that there exists some c>0c>0 such that h(n)∼c(nlog⁡n)1/2h(n) \sim c \left(\frac{n}{\log n}\right)^{1/2} as n→∞n\to \infty.Source says openFormalized