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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

79 Problems

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NumberQuestionOpen
#114No statement retained — open to read what the source holdsfalsifiableNo formal declaration
#115If p(z)p(z) is a polynomial of degree nn such that {z:∣p(z)∣≤1}\{z : \lvert p(z)\rvert\leq 1\} is connected then is it true that max⁡z∈C∣p(z)∣≤1∣p′(z)∣≤(12+o(1))n2?\max_{\substack{z\in\mathbb{C}\\ \lvert p(z)\rvert\leq 1}} \lvert p'(z)\rvert \leq (\tfrac{1}{2}+o(1))n^2?proved (Lean)Formalized
#116No statement retained — open to read what the source holdsprovedNo formal declaration
#119Is it true that lim sup⁡Mn=∞\limsup M_n = \infty?solvedFormalized
#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
#256No statement retained — open to read what the source holdsopenNo formal declaration
#395No statement retained — open to read what the source holdsprovedNo formal declaration
#485No statement retained — open to read what the source holdsprovedNo formal declaration
#494Selfridge and Straus [SeSt58] proved that AA is determined by AkA_k if ∣A∣|A| is divisible by a prime greater than kk.provedFormalized
#498Let z1,…,zn∈Cz_1,\ldots,z_n\in\mathbb{C} with 1≤∣zi∣1\leq \lvert z_i\rvert for 1≤i≤n1\leq i\leq n. Let DD be an arbitrary disc of radius 11. Is it true that the number of sums of the shape ∑i=1nϵizi for ϵi∈{−1,1}\sum_{i=1}^n\epsilon_iz_i \textrm{ for }\epsilon_i\in \{-1,1\} which lie in DD is at most (n⌊n/2⌋)\binom{n}{\lfloor n/2\rfloor}?proved (Lean)Formalized
#509Let f(z)∈C[z]f(z) ∈ ℂ[z] be a monic non-constant polynomial. Can the set {z∈C:∣f(z)∣≤1}\{z ∈ ℂ : |f(z)| ≤ 1\} be covered by a set of closed discs the sum of whose radii is ≤2≤ 2?openFormalized
#510Chowla's cosine problemopenFormalized
#511No statement retained — open to read what the source holdsdisprovedNo formal declaration
#512Is it true that, if A⊂ZA\subset \mathbb{Z} is a finite set of size NN, then ∫01∣∑n∈Ae(nθ)∣dθ≫log⁡N,\int_0^1 \left\lvert \sum_{n\in A}e(n\theta)\right\rvert \mathrm{d}\theta \gg \log N, where e(x)=e2πixe(x)=e^{2\pi ix }?proved (Lean)Formalized
#513Let f be a transcendental entire function. What is the greatest possible value of liminf (fun r : ℝ => ratio r f) atTop?openFormalized
#514No statement retained — open to read what the source holdsopenNo formal declaration
#515No statement retained — open to read what the source holdsprovedNo formal declaration
#516Let f = ∑ aₖzⁿₖ be an entire function of finite order such that nₖ / k → ∞. Then limsup (fun r => ratio r f) atTop = 1. This is proved in [Fu63].provedFormalized
#517If f(z) = ∑ aₖzⁿₖ is an entire function (with aₖ ≠ 0 for all k) such that nₖ / k → ∞, is it true that f assumes every value infinitely often?openFormalized
#519Let z1,…,zn∈Cz_1,\ldots,z_n\in \mathbb{C} with z1=1z_1=1. Must there exist an absolute constant c>0c>0 such that max⁡1≤k≤n∣∑izik∣>c? \max_{1\leq k\leq n}\left\lvert \sum_{i}z_i^k\right\rvert>c? proved (Lean)Formalized
#521Let (ϵk)k≥0(\epsilon_k)_{k\geq 0} be independently uniformly chosen at random from {−1,1}\{-1,1\}. If RnR_n counts the number of real roots of fn(z)=∑0≤k≤nϵkzkf_n(z)=\sum_{0\leq k\leq n}\epsilon_k z^k then is it true that, almost surely, lim⁡n→∞Rnlog⁡n=2π?\lim_{n\to \infty}\frac{R_n}{\log n}=\frac{2}{\pi}?openFormalized
#522Let f(z)=∑0≤k≤nϵkzkf(z)=\sum_{0\leq k\leq n} \epsilon_k z^k be a random polynomial, where ϵk∈{−1,1}\epsilon_k\in \{-1,1\} independently uniformly at random for 0≤k≤n0\leq k\leq n.openFormalized
#523No statement retained — open to read what the source holdsprovedNo formal declaration
#524No statement retained — open to read what the source holdsopenNo formal declaration
#525No statement retained — open to read what the source holdsprovedNo formal declaration
#527No statement retained — open to read what the source holdsprovedNo formal declaration
#671No statement retained — open to read what the source holdsopenNo formal declaration
#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.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))?proved (Lean)Formalized
#908No statement retained — open to read what the source holdsprovedNo formal declaration
#909No statement retained — open to read what the source holdsprovedNo formal declaration
#967Let 1<a1<⋯1<a_1<\cdots be a sequence of integers such that ∑1ai<∞\sum\frac{1}{a_i}<\infty. Is it true that, for every t∈Rt\in \mathbb{R}, 1+∑k1ak1+it≠0?1+\sum_{k}\frac{1}{a_k^{1+it}}\neq 0?disproved (Lean)Formalized
#973Does there exist a constant C>1C>1 such that, for every n≥2n\geq 2, there exists a sequence zi∈Cz_i\in \mathbb{C} with z1=1z_1=1 and ∣zi∣≥1\lvert z_i\rvert \geq 1 for all 1≤i≤n1\leq i\leq n with max⁡2≤k≤n+1∣∑1≤i≤nzik∣<C−n\max_{2\leq k\leq n+1}\left\lvert \sum_{1\leq i\leq n}z_i^k\right\rvert < C^{-n}?openFormalized
#974Let z1,…,zn∈Cz_1,\ldots,z_n\in \mathbb{C} be a sequence such that z1=1z_1=1. Suppose that the sequence of sk=∑1≤i≤nziks_k=\sum_{1\leq i\leq n}z_i^k contains infinitely many (n−1)(n-1)-tuples of consecutive values of sks_k which are all 00. Then (essentially) zj=e(j/n),z_j=e(j/n), where e(x)=e2πixe(x)=e^{2\pi ix}.proved (Lean)Formalized
#987Question 1:provedFormalized
#990Let f=a0+⋯+adxd∈C[x]f=a_0+\cdots+a_dx^d\in \mathbb{C}[x] be a polynomial. Is it true that, if ff has roots z1,…,zdz_1,\ldots,z_d with corresponding arguments θ1,…,θd∈[0,2π]\theta_1,\ldots,\theta_d\in [0,2\pi], then for all intervals I⊆[0,2π]I\subseteq [0,2\pi] ∣(#θi∈I)−∣I∣2πd∣≪(nlog⁡M)1/2, \left\lvert (\# \theta_i \in I) - \frac{\lvert I\rvert}{2\pi}d\right\rvert \ll \left(n\log M\right)^{1/2}, where nn is the number of non-zero coefficients of ff and M=∣a0∣+⋯+∣ad∣(∣a0∣∣ad∣)1/2. M=\frac{\lvert a_0\rvert+\cdots +\lvert a_d\rvert}{(\lvert a_0\rvert\lvert a_d\rvert)^{1/2}}. disproved (Lean)Formalized
#994No statement retained — open to read what the source holdsdisprovedNo formal declaration
#995No statement retained — open to read what the source holdsopenNo formal declaration
#996Does there exists a positive constant C such that for all f ∈ L²[0,1] and all lacunary sequences n, if ‖f - fₖ‖₂ = O(1 / log log log k ^ C), then for almost every x, lim ∑ k ∈ Finset.range N, f (n k • x)) / N = ∫ t, f t ∂t?openFormalized
#997Is it true that, for every α\alpha, the sequence {αpn}\{ \alpha p_n\} is not well-distributed, if pnp_n is the sequence of primes?proved (Lean)Formalized
#998No statement retained — open to read what the source holdsprovedNo formal declaration
#1002For any 0<α<10<\alpha<1, let f(α,n)=1log⁡n∑1≤k≤n(12−{αk})f(\alpha,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}(\tfrac{1}{2}- \{ \alpha k\}). Does f(α,n)f(\alpha,n) have an asymptotic distribution function?openFormalized
#1038What is the infimum of |{x ∈ ℝ : |f x| < 1}| over all nonconstant monic polynomials f such that all of its roots are real and contained in [-1,1]?openFormalized

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