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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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576 Problems · 1 with reviewed evidence

10/12

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#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
#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
#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
#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
#913Are there infinitely many nn such that if n(n+1)=∏ipiki n(n + 1) = \prod_i p_i^{k_i} is the factorisation into distinct primes then all exponents kik_i are distinct?Source says openFormalized
#928No statement retained — open to read what the source holdsSource says openNo formal declaration
#929No statement retained — open to read what the source holdsSource says openNo formal declaration
#930Is it true that, for every rr, there is a kk such that if I1,…,IrI_1,\ldots,I_r are disjoint intervals of consecutive integers, all of length at least kk, then ∏1≤i≤r∏m∈Iim \prod_{1\leq i\leq r}\prod_{m\in I_i}m is not a perfect power?Source says openFormalized
#931Let k1≥k2≥3k_1 \geq k_2 \geq 3. Are there only finitely many n2≥n1+k1n_2\geq n_1 + k_1 such that ∏1≤i≤k1(n1+i) and ∏1≤j≤k2(n2+j) \prod_{1\leq i\leq k_1}(n_1 + i)\ \text{and}\ \prod_{1\leq j\leq k_2} (n_2 + j) have the same prime factors?Source says openFormalized
#932Let pkp_k denote the kkth prime. For infinitely many rr there are at least two integers pr<n<pr+1p_r < n < p_{r+1} all of whose prime factors are <pr+1−pr< p_{r + 1} - p_r.Source says openFormalized
#933If n(n+1)=2k3lmn(n+1)=2^k3^lm, where (m,6)=1(m,6)=1, then is it true that lim sup⁡n→∞2k3lnlog⁡n=∞\limsup_{n\to \infty} \frac{2^k3^l}{n\log n}=\infty?Source says openFormalized
#935No statement retained — open to read what the source holdsSource says openNo formal declaration
#936Is n!+1n! + 1 powerful for finitely many nn?Source says openFormalized
#937Are there infinitely many four-term arithmetic progressions of coprime powerful numbers? (A number nn is *powerful* if p∣n→p2∣np \mid n \to p^2 \mid n; Nat.Powerful.)Source says provedFormalized
#938Let A={n1<n2<⋯ }A=\{n_1 < n_2 < \cdots\} be the sequence of powerful numbers (if p∣np\mid n then p2∣np^2\mid n). Are there only finitely many three-term progressions of consecutive terms nk,nk+1,nk+2n_k,n_{k+1},n_{k+2}?Source says openFormalized
#939If r≥4r≥4 then can the sum of r−2r-2 coprime rr-powerful numbers ever be itself rr-powerful?Source says openFormalized
#940Let r≥3r \ge 3. Is it true that the set of integers which are the sum of at most rr rr-powerful numbers has density 00?Source says openFormalized
#941No statement retained — open to read what the source holdsSource says provedNo formal declaration
#942Is there some constant c>0c > 0 such that h(n)<(log⁡n)c+o(1)h(n) < (\log n)^{c + o(1)} and, for infinitely many nn, h(n)>(log⁡n)c−o(1)h(n) > (\log n)^{c - o(1)}.Source says openFormalized
#943Let AA be the set of powerful numbers. Is is true that 1A∗1A(n)=no(1)1_A\ast 1_A(n)=n^{o(1)} for every nn?Source says openFormalized
#945Is it true that F(x)≤(log⁡x)O(1)F(x) \leq (\log x)^{O(1)}?Source says openFormalized
#946There are infinitely many nn such that τ(n)=τ(n+1)τ(n) = τ(n+1). Proved in [He84]. Here τ is the divisor counting function, which is σ 0 in mathlib.Source says provedFormalized
#947No statement retained — open to read what the source holdsSource says proved (Lean)No formal declaration
#948No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#950Is it true that lim inf⁡f(n)=1\liminf f(n)=1?Source says openFormalized
#951If 1 < a 0 < ... has property Erdos951Prop, is it true that #{a i ≤ x} ≤ π x?Source says openFormalized
#952Is there an infinite sequence of distinct Gaussian primes x1,x2,…x_1,x_2,\ldots such that ∣xn+1−xn∣≪1\lvert x_{n+1}-x_n\rvert \ll 1?Source says openFormalized
#954No statement retained — open to read what the source holdsSource says openNo formal declaration
#955If A⊂NA\subset \mathbb{N} has density 00 then s−1(A)s^{-1}(A) must also have density 00.Source says openFormalized
#961It is conjectured that f(k)≪(log⁡k)O(1)f(k) \ll (\log k)^O(1).Source says openFormalized