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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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NumberQuestionOpen
#961It is conjectured that f(k)≪(log⁡k)O(1)f(k) \ll (\log k)^O(1).openFormalized
#962Main conjecture:openFormalized
#963No statement retained — open to read what the source holdsopenNo formal declaration
#964No statement retained — open to read what the source holdsproved (Lean)No formal declaration
#965Erdős asks in [Er75b] if for every 2-coloring of ℝ, there is an uncountable set A⊆RA ⊆ ℝ such that all sums a+ba + b for a,b∈A,a≠ba, b ∈ A, a ≠ b have the same colour.disprovedFormalized
#966Let k,r≥2k,r\geq 2. Does there exist a set A⊆NA\subseteq \mathbb{N} that contains no non-trivial arithmetic progression of length k+1k+1, yet in any rr-colouring of AA there must exist a monochromatic non-trivial arithmetic progression of length kk?proved (Lean)Formalized
#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
#968Does the set {n | u n < u (n+1)} have positive lower density?openFormalized
#969No statement retained — open to read what the source holdsopenNo formal declaration
#970No statement retained — open to read what the source holdsopenNo formal declaration
#971Let p(a, d) be the least prime congruent to a (mod d). Does there exist a constant c > 0 such that for all large d, p(a, d) > (1 + c) * φ(d) * log d for ≫ φ(d) many values of a?openFormalized
#972Erdős problem 972. Let α>1\alpha > 1 be irrational. Are there infinitely many primes pp such that ⌊pα⌋\lfloor p\alpha \rfloor is also prime?openFormalized
#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
#975For an irreducible polynomial f∈Z[x]f \in \mathbb{Z}[x] with f(n)≥1f(n) \ge 1 for sufficiently large nn, does there exists a constant c=c(f)>0c = c(f) > 0 such that ∑n≤xτ(f(n))≈c⋅xlog⁡x\sum_{n \le x} \tau(f(n)) \approx c \cdot x \log x?openFormalized
#976No statement retained — open to read what the source holdsopenNo formal declaration
#977No statement retained — open to read what the source holdsprovedNo formal declaration
#978Let f ∈ ℤ[X] be an irreducible polynomial with positive leading coefficient. Suppose that the degree k of f is larger than 2, is not equal to a power of 2, and f n has no fixed (k - 1)-th power divisors other than 1. Then the set of n such that f n is (k - 1)-th power free has positive density, and this is proved in [Ho67].openFormalized
#979Let k≥2k ≥ 2, and let fk(n)f_k(n) count the number of solutions to n=p1k+⋯+pkkn = p_1^k + \dots + p_k^k, where the pip_i are prime numbers. Is it true that lim sup⁡fk(n)=∞\limsup f_k(n) = \infty?openFormalized
#980No statement retained — open to read what the source holdsprovedNo formal declaration
#981No statement retained — open to read what the source holdsprovedNo formal declaration
#982If nn distinct points in R2\mathbb{R}^2 form a convex polygon then some vertex has at least ⌊n2⌋\lfloor\frac{n}{2}\rfloor different distances to other vertices.falsifiableFormalized
#983No statement retained — open to read what the source holdsopenNo formal declaration
#984No statement retained — open to read what the source holdsprovedNo formal declaration
#985Is it true that, for every prime pp, there is a prime q≤pq \leq p which is a primitive root modulo pp?openFormalized
#986No statement retained — open to read what the source holdsprovedNo formal declaration
#987Question 1:provedFormalized
#988No statement retained — open to read what the source holdssolvedNo formal declaration
#989No statement retained — open to read what the source holdssolvedNo formal declaration
#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
#991No statement retained — open to read what the source holdsprovedNo formal declaration
#992No statement retained — open to read what the source holdsdisprovedNo formal declaration
#993No statement retained — open to read what the source holdsfalsifiableNo formal declaration
#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
#999No statement retained — open to read what the source holdsprovedNo formal declaration
#1000Let A={n1<n2<⋯ }A=\{n_1<n_2<\cdots\} be an infinite sequence of integers, and let ϕA(k)\phi_A(k) count the number of 1≤m≤nk1\leq m\leq n_k such that the fraction mnk\frac{m}{n_k} does not have denominator njn_j for j<kj<k when written in lowest form; equivalently, nk(m,nk)≠nj \frac{n_k}{(m,n_k)}\neq n_j for all 1≤j<k1\leq j<k.proved (Lean)Formalized
#1001No statement retained — open to read what the source holdssolvedNo 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
#1003Are there infinitely many solutions to ϕ(n)=ϕ(n+1)\phi(n) = \phi(n+1), where ϕ\phi is the Euler totient function?openFormalized
#1004For any fixed c > 0, if x is sufficiently large then there exists n ≤ x such that the values of φ(n+k) are all distinct for 1 ≤ k ≤ (log x)^c. This is an open problem.openFormalized
#1005No statement retained — open to read what the source holdsopenNo formal declaration
#1006No statement retained — open to read what the source holdsdisprovedNo formal declaration
#1007The dimension of a graph GG is the minimal nn such that GG can be embedded in Rn\mathbb{R}^n such that every edge of GG is a unit line segment.solved (Lean)Formalized
#1008Does every graph with mm edges contain a subgraph with ≫m2/3\gg m^{2/3} edges which contains no C4C_4?proved (Lean)Formalized

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