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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
  • Other source status53
Exact formal statement available604 / 1,217
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Problems

1,217 Problems · 2 with reviewed evidence

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NumberQuestionOpen
#193Let S⊆Z3S \subseteq \mathbb{Z}^3 be a finite set and let A={a1,a2,…}A = \lbrace a_1, a_2, \ldots \rbrace be an infinite SS-walk, so that ai+1−ai∈Sa_{i+1} - a_i \in S for all ii. Must AA contain three collinear points?openFormalized
#194Let k≥3k\geq 3. Must any ordering of R\mathbb{R} contain a monotone kk-term arithmetic progression, that is, some x1<⋯<xkx_1 <\cdots < x_k which forms an increasing or decreasing kk-term arithmetic progression?disproved (Lean)Formalized
#195What is the largest kk such that in any permutation of Z\mathbb{Z} there must exist a monotone kk-term arithmetic progression x1<⋯<xkx_1 < \cdots < x_k?openFormalized
#196Must every permutation of N\mathbb{N}, contain a monotone 4-term arithmetic progression?openFormalized
#197Can N\mathbb{N} be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithmetic progressions?openFormalized
#198The statement for which Baumgartner actually writes a proof.disproved (Lean)Formalized
#199If A⊂RA\subset \mathbb{R} does not contain a 3-term arithmetic progression then must R\A\mathbb{R}\backslash A contain an infinite arithmetic progression?disproved (Lean)Formalized
#200Does the longest arithmetic progression of primes in {1,…,N}\{1,\ldots,N\} have length o(log⁡N)o(\log N)?openFormalized
#201No statement retained — open to read what the source holdsopenNo formal declaration
#202Let n1<⋯<nr≤Nn_1<\cdots < n_r\leq N with associated ai(modni)a_i\pmod{n_i} such that the congruence classes are disjoint (that is, every integer is ≡ai(modni)\equiv a_i\pmod{n_i} for at most one 1≤i≤r1\leq i\leq r). How large can rr be in terms of NN?solved (Lean)Formalized
#203Is there an integer mm with (m,6)=1(m, 6) = 1 such that none of 2k⋅3ℓ⋅m+12^k \cdot 3^\ell \cdot m + 1 are prime, for any k,ℓ≥0k, \ell \ge 0?openFormalized
#204Are there nn such that there is a covering system with moduli the divisors of nn which is 'as disjoint as possible'?disproved (Lean)Formalized
#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.)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.disproved (Lean)Formalized
#207No statement retained — open to read what the source holdsprovedNo formal declaration
#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?openFormalized
#209Let AA be a finite collection of d≥4d\geq 4 non-parallel lines in R2\mathbb{R}^2 such that there are no points where at least four lines from AA meet. Must there exist a 'Gallai triangle' (or 'ordinary triangle'): three lines from AA which intersect in three points, and each of these intersection points only intersects two lines from AA?disproved (Lean)Formalized
#210No statement retained — open to read what the source holdsprovedNo formal declaration
#211No statement retained — open to read what the source holdsprovedNo formal declaration
#212Is there a dense subset of ℝ^2 such that all pairwise distances are rational?openFormalized
#213Let n≥4n \geq 4. Are there nn points in R2\mathbb{R}^2, no three on a line and no four on a circle, such that all pairwise distances are integers?openFormalized
#214Let S⊂R2S\subset \mathbb{R}^2 be such that no two points in SS are distance 11 apart. Must the complement of SS contain four points which form a unit square?proved (Lean)Formalized
#215No statement retained — open to read what the source holdsprovedNo formal declaration
#216No statement retained — open to read what the source holdsdisprovedNo formal declaration
#217No statement retained — open to read what the source holdsopenNo formal declaration
#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.openFormalized
#219Are there arbitrarily long arithmetic progressions of primes? Solution: yes. Ref: Green, Ben and Tao, Terence, _The primes contain arbitrarily long arithmetic progressions_provedFormalized
#220No statement retained — open to read what the source holdsprovedNo 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?proved (Lean)Formalized
#222No statement retained — open to read what the source holdsopenNo formal declaration
#223No statement retained — open to read what the source holdssolvedNo formal declaration
#224If A⊆RdA\subseteq \mathbb{R}^d is any set of 2d+12^d+1 points then some three points in AA determine an obtuse angle.proved (Lean)Formalized
#225No statement retained — open to read what the source holdsprovedNo formal declaration
#226Is there an entire non-linear function ff such that, for all x∈Rx\in\mathbb{R}, xx is rational if and only if f(x)f(x) is?proved (Lean)Formalized
#227No statement retained — open to read what the source holdsdisprovedNo formal declaration
#228Does there exist, for all large nn, a polynomial PP of degree nn, with coefficients ±1\pm1, such that n≪∣P(z)∣≪n\sqrt n \ll |P(z)| \ll \sqrt n for all ∣z∣=1|z|=1, with the implied constants independent of zz and nn?provedFormalized
#229Let (Sn)n≥1(S_n)_{n \ge 1} be a sequence of sets of complex numbers, none of which have a finite limit point. Does there exist an entire transcendental function f(z)f(z) such that, for all n≥1n \ge 1, there exists some kn≥0k_n \ge 0 such that f(kn)(z)=0f^{(k_n)}(z) = 0 for all z∈Snz \in S_n.proved (Lean)Formalized
#230No statement retained — open to read what the source holdsdisprovedNo formal declaration
#231No statement retained — open to read what the source holdsdisproved (Lean)No formal declaration
#232No statement retained — open to read what the source holdsprovedNo 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.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?openFormalized
#235No statement retained — open to read what the source holdsprovedNo 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).openFormalized
#237No statement retained — open to read what the source holdsproved (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?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?provedFormalized
#240No statement retained — open to read what the source holdsprovedNo formal declaration

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