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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
  • Resolved per source556
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Problems

576 Problems · 1 with reviewed evidence

2/12

SelectNumberQuestionOpen
#55No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#56Suppose A⊆{1,…,N}A \subseteq \{1,\dots,N\} is such that there are no k+1k+1 elements of AA which are relatively prime. An example is the set of all multiples of the first kk primes. Is this the largest such set? To avoid trivial counterexamples, we must insist that NN be at least the kkth prime.Source says disproved (Lean)Formalized
#66Is there and A⊂NA \subset \mathbb{N} is such that lim⁡n→∞1A∗1A(n)log⁡n\lim_{n\to \infty}\frac{1_A\ast 1_A(n)}{\log n} exists and is ≠0\ne 0?Source says openFormalized
#68Is ∑n=2∞1n!−1\sum_{n=2}^\infty \frac{1}{n!-1} irrational?Source says openFormalized
#69Is ∑n≥2ω(n)2n \sum_{n\geq 2}\frac{\omega(n)}{2^n} irrational? (Here ω(n)\omega(n) counts the number of distinct prime divisors of nn.)Source says provedFormalized
#121No statement retained — open to read what the source holdsSource says disprovedNo formal declaration
#122No statement retained — open to read what the source holdsSource says openNo formal declaration
#123Let a,b,ca, b, c be three integers which are pairwise coprime. Is every large integer the sum of distinct integers of the form akblcma^k b^l c^m (k,l,m≥0k, l, m ≥ 0), none of which divide any other?Source says proved (Lean)Formalized
#124Let 3≤d1<d2<⋯<dr3\leq d_1 < d_2 < \cdots < d_r be integers such that all sufficiently large integers can be written as a sum of the shape ∑iciai\sum_i c_ia_i where ci∈{0,1}c_i \in \{0, 1\} and aia_i has only the digits 0,10, 1 when written in base did_i. Then ∑1≤i≤r1di−1≥1.\sum_{1 \le i \le r}\frac 1{d_i - 1} \ge 1.Source says openFormalized
#125Case 3: Does A+BA + B have positive upper and lower density that are equal? This is the literal interpretation of "positive density" which was falsified.Source says disproved (Lean)Formalized
#126Let f(n)f(n) be maximal such that if A⊆NA\subseteq\mathbb{N} has ∣A∣=n|A| = n then ∏a≠b∈A(a+b)\prod_{a\neq b\in A}(a + b) has at least f(n)f(n) distinct prime factors. Is it true that f(n)log⁡n→∞\frac{f(n)}{\log n} \to\infty?Source says openFormalized
#131No statement retained — open to read what the source holdsSource says openNo formal declaration
#137Let k≥3k\geq 3. Can the product of any kk consecutive integers NN ever be powerful? That is, must there always exist a prime p∣Np\mid N such that p2∤Np^2\nmid N?Source says openFormalized
#144No statement retained — open to read what the source holdsSource says provedNo formal declaration
#145Let s1<s2<⋯s_1 < s_2 < \cdots be the sequence of squarefree numbers. Is it true that, for any α≥0\alpha\geq 0, lim⁡x→∞1x∑sn≤x(sn+1−sn)α \lim_{x\to\infty} \frac{1}{x}\sum_{s_n\leq x}(s_{n+1}-s_n)^\alpha exists?Source says openFormalized
#148No statement retained — open to read what the source holdsSource says openNo formal declaration
#164A set A⊂NA\subset \mathbb{N} is primitive if no member of AA divides another. Is the sum ∑n∈A1nlog⁡n\sum_{n\in A}\frac{1}{n\log n} maximised over all primitive sets when AA is the set of primes?Source says proved (Lean)Formalized
#175No statement retained — open to read what the source holdsSource says provedNo formal declaration
#205Is it true that all sufficiently large nn can be written as 2k+m2^k+m for some k≥0k\geq 0, where Ω(m)<log⁡log⁡m\Omega(m)<\log\log m? (Here Ω(m)\Omega(m) is the number of prime divisors of mm counted with multiplicity.)Source says disproved (Lean)Formalized
#206Let x>0x>0 be a real number. For any n≥1n\geq 1 let Rn(x)=∑i=1n1mi<xR_n(x) = \sum_{i=1}^n\frac{1}{m_i}<x be the maximal sum of nn distinct unit fractions which is <x<x.Source says disproved (Lean)Formalized
#208Let s1<s2<…s_1 < s_2 < \dots be the sequence of squarefree numbers. Is it true that for any ϵ>0\epsilon > 0 and large nn, sn+1−sn≪ϵsnϵs_{n+1} - s_n \ll_\epsilon s_n^\epsilon?Source says openFormalized
#218The set of indices nn for which a prime gap is preceded by a larger or equal prime gap has a natural density of 12\frac 1 2.Source says openFormalized
#219Are 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
#220No statement retained — open to read what the source holdsSource says provedNo formal declaration
#221Is there a set A⊂NA\subset\mathbb{N} such that, for all large NN, ∣A∩{1,…,N}∣≪N/log⁡N\lvert A\cap\{1,\ldots,N\}\rvert \ll N/\log N and such that every large integer can be written as 2k+a2^k+a for some k≥0k\geq 0 and a∈Aa\in A?Source says proved (Lean)Formalized
#222No statement retained — open to read what the source holdsSource says openNo formal declaration
#233A conjecture by Heath-Brown: The sum of squares of the first NN gaps between consecutive primes behaves like N∗(logN)2N * (log N)^2.Source says openFormalized
#234Is 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
#235No statement retained — open to read what the source holdsSource says provedNo formal declaration
#236Let f(n)f(n) count the number of solutions to n=p+2kn=p+2^k for prime pp and k≥0k\geq 0. Show that f(n)=o(log⁡n)f(n)=o(\log n).Source says openFormalized
#237No statement retained — open to read what the source holdsSource says proved (Lean)No formal declaration
#238Let 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
#239Let f:N→{−1,1}f:\mathbb{N}\to \{-1,1\} be a multiplicative function. Is it true that lim⁡N→∞1N∑n≤Nf(n) \lim_{N\to \infty}\frac{1}{N}\sum_{n\leq N}f(n) always exists?Source says provedFormalized
#240No statement retained — open to read what the source holdsSource says provedNo formal declaration
#242For every n>2n>2 there exist distinct integers 1≤x<y<z1 ≤ x < y < z such that 4n=1x+1y+1z\frac 4 n = \frac 1 x + \frac 1 y + \frac 1 z.Source says falsifiableFormalized
#243Let a1<a2<…a_1 < a_2 < \dots be a sequence of integers such that lim⁡n→∞anan−12=1\lim_{n\to\infty} \frac{a_n}{a_{n-1}^2} = 1 and ∑1an∈Q\sum \frac{1}{a_n} \in \mathbb{Q}.Source says openFormalized
#244Let C>1C > 1. Does the set of integers of the form p+⌊Ck⌋p + \lfloor C^k \rfloor, for some prime pp and k≥0k\geq 0, have density >0>0?Source says openFormalized
#246Let (a,b)=1(a,b)=1. The set {akbl:k,l≥0}\{a^kb^l: k,l\geq 0\} is complete - that is, every large integer is the sum of distinct integers of the form akbla^kb^l with k,l≥0k,l\geq 0.Source says proved (Lean)Formalized
#247Let n1<n2<⋯n_1 < n_2 < \cdots be a sequence of integers such that lim sup⁡nkk=∞. \limsup \frac{n_k}{k} = \infty. Is ∑k=1∞12nk \sum_{k=1}^{\infty} \frac{1}{2^{n_k}} transcendental?Source says openFormalized
#248Are there infinitely many nn such that ω(n+k)≪k\omega(n + k) \ll k for all k≥1k \geq 1? Here ω(n)\omega(n) is the number of distinct prime divisors of nn.Source says provedFormalized
#249Is ∑nϕ(n)2n\sum_{n} \frac{\phi(n)}{2^n} irrational? Here ϕ\phi is the Euler totient function.Source says openFormalized
#250Is ∑n=1∞σ(n)2n \sum_{n=1}^\infty \frac{\sigma(n)}{2^n} irrational? Here σ(n)\sigma(n) is the sum of divisors function.Source says provedFormalized
#251Is ∑n=1∞pn2n\sum_{n=1}^\infty \frac{p_n}{2^n} irrational? Here pnp_n is the nn-th prime (p1=2,p2=3,…p_1=2, p_2=3, \dots).Source says openFormalized
#252Erdős Problem 252: irrationality of the sum for a given kk.Source says openFormalized
#253Let a1<a2<…a_1 < a_2 < \dotsc be an infinite sequence of positive integers such that ai+1ai→1\frac{a_{i+1}}{a_i} \to 1. If every arithmetic progression contains infinitely many integers which are the sum of distinct aia_i then every sufficiently large integer is the sum of distinct aia_i.Source says disprovedFormalized
#254Let A⊆NA\subseteq \mathbb{N} be such that ∣A∩[1,2x]∣−∣A∩[1,x]∣→∞ as x→∞\lvert A\cap [1,2x]\rvert -\lvert A\cap [1,x]\rvert \to \infty\textrm{ as }x\to \infty and ∑n∈A{θn}=∞\sum_{n\in A} \{ \theta n\}=\infty for every θ∈(0,1)\theta\in (0,1), where {x}\{x\} is the distance of xx from the nearest integer. Then every sufficiently large integer is the sum of distinct elements of AA.Source says openFormalized
#261No statement retained — open to read what the source holdsSource says openNo formal declaration
#268Let X be the set of points in Fin d → ℝ of the shape fun i : Fin d => ∑' n : A, (1 : ℝ) / (n + i) for some infinite subset A ⊆ ℕ such that 1 / n is summable over A. X has nonempty interior. This is proved in [KoTa24]. -Source says proved (Lean)Formalized