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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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#817Let k≥3k \geq 3. Define gk(n)g_k(n) to be the minimal NN such that {1,...,N}\{1, ..., N\} contains some AA of size ∣A∣=n|A| = n such that ⟨A⟩={∑a∈Aϵaa:ϵa∈{0,1}} \langle A\rangle = \left\{\sum_{a \in A} \epsilon_a a : \epsilon_a \in\{0, 1\}\right\} contains no non-trivial kk-term arithmetic progression. Estimate gk(n)g_k(n). In particular, is it true that g3(n)≫3n g_3(n) \gg 3^n Source says openFormalized
#818Let AA be a finite set of integers such that ∣A+A∣≪∣A∣\lvert A+A\rvert \ll \lvert A\rvert. Is it true that ∣AA∣≫∣A∣2(log⁡∣A∣)C\lvert AA\rvert \gg \frac{\lvert A\rvert^2}{(\log \lvert A\rvert)^C} for some constant C>0C>0?Source says proved (Lean)Formalized
#819No statement retained — open to read what the source holdsSource says openNo formal declaration
#820No statement retained — open to read what the source holdsSource says openNo formal declaration
#821Is it true that, for every ϵ>0\epsilon>0, there exist infinitely many nn such that g(n)>n1−ϵg(n) > n^{1-\epsilon}?Source says openFormalized
#822Does the set of integers of the form n+φ(n)n + \varphi(n) have positive (lower) density?Source says provedFormalized
#823No statement retained — open to read what the source holdsSource says provedNo formal declaration
#824No statement retained — open to read what the source holdsSource says openNo formal declaration
#825Is there an absolute constant C>0C > 0 such that every integer nn with σ(n)>Cn\sigma(n) > Cn is the distinct sum of proper divisors of nn?Source says provedFormalized
#826Are there infinitely many nn such that, for all k≥1k\geq 1 τ(n+k)≪k? \tau(n + k) \ll k? Source says openFormalized
#827No statement retained — open to read what the source holdsSource says openNo formal declaration
#828Is it true that, for any a∈Za \in \mathbb{Z}, there are infinitely many nn such that ϕ(n)∣n+a\phi(n) | n + a?Source says openFormalized
#829Erdős Problem 829 (open). Let A⊆NA \subseteq \mathbb{N} be the set of perfect cubes. Is it true that (1A∗1A)(n)≪(log⁡n)O(1)(1_A \ast 1_A)(n) \ll (\log n)^{O(1)}? That is, does there exist a natural number CC such that the number of representations of nn as a sum of two cubes is O((log⁡n)C)O((\log n)^C) as n→∞n \to \infty?Source says openFormalized
#830Erdos Problem 830, Part 1 We say that a,b∈Na,b\in \mathbb{N} are an amicable pair if σ(a)=σ(b)=a+b\sigma(a)=\sigma(b)=a+b. Are there infinitely many amicable pairs?Source says openFormalized
#831No statement retained — open to read what the source holdsSource says openNo formal declaration
#832No statement retained — open to read what the source holdsSource says disprovedNo formal declaration
#833No statement retained — open to read what the source holdsSource says provedNo formal declaration
#834No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#835It is known that for 3≤k≤83 \leq k \leq 8, the chromatic number of J(2k,k)J(2k, k) is greater than k+1k+1, see Johnson graphs.Source says verifiableFormalized
#836No statement retained — open to read what the source holdsSource says openNo formal declaration
#837No statement retained — open to read what the source holdsSource says openNo formal declaration
#838No statement retained — open to read what the source holdsSource says openNo formal declaration
#839No statement retained — open to read what the source holdsSource says openNo formal declaration
#840No statement retained — open to read what the source holdsSource says openNo formal declaration
#841No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#842No statement retained — open to read what the source holdsSource says provedNo formal declaration
#843No statement retained — open to read what the source holdsSource says provedNo formal declaration
#844Let A⊆{1,…,N}A\subseteq \{1,\ldots,N\} be such that, for all a,b∈Aa,b\in A, the product abab is not squarefree.Source says proved (Lean)Formalized
#845Let C>0C > 0. Is it true that the set of integers of the form n=b1+⋯+btn = b_1 + \cdots + b_t, with b1<⋯<btb_1 < \cdots < b_t, where bi=2ki3lib_i = 2^{k_i}3^{l_i} for 1≤i≤t1 \leq i\leq t and bt≤Cb1b_t \leq Cb_1 has density 00?Source says disproved (Lean)Formalized
#846Erdős Problem 846 Let A ⊂ ℝ² be an infinite set for which there exists some ϵ>0 such that in any subset of A of size n there are always at least ϵn with no three on a line. Is it true that A is the union of a finite number of sets where no three are on a line?Source says disproved (Lean)Formalized
#847Let A⊂NA \subset \mathbb{N} be an infinite set for which there exists some ϵ>0\epsilon > 0 such that in any subset of AA of size nn there is a subset of size at least ϵn\epsilon n which contains no three-term arithmetic progression.Source says disprovedFormalized
#848Is the maximum size of a set A⊆{1,…,N}A ⊆ \{1, \dots, N\} such that ab+1ab + 1 is never squarefree (for all a,b∈Aa, b ∈ A) achieved by taking those n≡7(mod25)n ≡ 7 \pmod{25}?Source says decidableFormalized
#849Is it true that, for every integer t≥1t\geq1, there is some integer aa such that (nk)=a{n \choose k} = a with 1≤k≤n21\leq k \le \frac{n}{2} has exactly tt solutions?Source says openFormalized
#850Can there exist two distinct integers xx and yy such that x,yx,y have the same prime factors, x+1,y+1x+1,y+1 have the same prime factors, and x+2,y+2x+2,y+2 also have the same prime factors?Source says openFormalized
#851Let ϵ>0\epsilon > 0. Is there some r≪ϵ1r \ll_\epsilon 1 such that the density of integers of the form 2k+n2^k+n, where k≥0k \geq 0 and nn has at most rr prime divisors, is at least 1−ϵ1-\epsilon?Source says provedFormalized
#852No statement retained — open to read what the source holdsSource says openNo formal declaration
#853Let dn=pn+1−pnd_n = p_{n+1} - p_n, where pnp_n is the nnth prime. Let r(x)r(x) be the smallest even integer tt such that dn=td_n = t has no solutions for n≤xn \le x.Source says openFormalized
#854No statement retained — open to read what the source holdsSource says openNo formal declaration
#855Erdős Problem 855 (Segal's conjecture): π(x+y)≤π(x)+π(y)\pi(x + y) \le \pi(x) + \pi(y) for sufficiently large x,yx, y.Source says openFormalized
#856No statement retained — open to read what the source holdsSource says openNo formal declaration
#857Estimate m(n,k), or better give an asymptotic formula.Source says openFormalized
#858No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#859The density of the divisor sum set is asymptotically equivalent to c1/log⁡(t)c2c_1 / \log(t)^{c_2}.Source says openFormalized
#860No statement retained — open to read what the source holdsSource says openNo formal declaration
#861No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#862Let A1(N)A_1(N) be the number of maximal Sidon subsets of {1,…,N}\{1,\ldots,N\}. Is it true that A1(N)<2o(N1/2)?A_1(N) < 2^{o(N^{1/2})}?Source says solved (Lean)Formalized
#863No statement retained — open to read what the source holdsSource says provedNo formal declaration
#864No statement retained — open to read what the source holdsSource says openNo formal declaration