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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

31 Problems

SelectNumberQuestionOpen
#18Erdős's Theorem. Erdős proved that h(n!)<nh(n!) < n for all n≥1n \ge 1.Source says openFormalized
#26Let A⊂NA\subset\mathbb{N} be infinite such that ∑a∈A1a=∞\sum_{a \in A} \frac{1}{a} = \infty. Must there exist some k≥1k\geq 1 such that almost all integers have a divisor of the form a+ka+k for some a∈Aa\in A?Source says disproved (Lean)Formalized
#144No statement retained — open to read what the source holdsSource says provedNo formal declaration
#381No statement retained — open to read what the source holdsSource says disprovedNo formal declaration
#444No statement retained — open to read what the source holdsSource says provedNo formal declaration
#446No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#448Let τ(n)\tau(n) count the divisors of nn and τ+(n)\tau^+(n) count the number of kk such that nn has a divisor in [2k,2k+1)[2^k, 2^{k+1}). Is it true that, for all ϵ>0\epsilon > 0, τ+(n)<ϵ⋅τ(n) \tau^+(n) < \epsilon \cdot \tau(n) for almost all nn?Source says disprovedFormalized
#449No statement retained — open to read what the source holdsSource says disprovedNo formal declaration
#450How large must y=y(ϵ,n)y=y(\epsilon,n) be such that the number of integers in (x,x+y)(x,x+y) with a divisor in (n,2n)(n,2n) is at most ϵy\epsilon y?Source says openFormalized
#468No statement retained — open to read what the source holdsSource says openNo formal declaration
#469Let AA be the set of all nn such that n=d1+⋯+dkn = d_1 + ⋯ + d_k with did_i distinct proper divisors of nn, but this is not true for any m∣nm ∣ n with m<nm < n. Does: ∑n∈A1n \sum_{n ∈ A} \frac 1 n converge?Source says proved (Lean)Formalized
#470Are there any odd weird numbers?Source says openFormalized
#673No statement retained — open to read what the source holdsSource says provedNo formal declaration
#692Let δ1(n,m)\delta_1(n,m) be the density of the set of integers with exactly one divisor in (n,m)(n,m). Is δ1(n,m)\delta_1(n,m) unimodular for m>n+1m>n+1 (i.e. increases until some mm then decreases thereafter)?Source says disproved (Lean)Formalized
#693No statement retained — open to read what the source holdsSource says openNo formal declaration
#696No statement retained — open to read what the source holdsSource says solved (Lean)No formal declaration
#697For each mm and α\alpha, the density of the set of integers which are divisible by some d≡1(modm)d \equiv 1 \pmod{m} with 1<d<exp⁡(mα)1 < d < \exp (m ^ \alpha) exists.Source says provedFormalized
#859The density of the divisor sum set is asymptotically equivalent to c1/log⁡(t)c2c_1 / \log(t)^{c_2}.Source says openFormalized
#884For a natural number n, let 1=d1<…<dτ(n)=n1 = d_1 < \dotsc < d_{\tau(n)} = n denote the divisors of n in increasing order. Does it hold that ∑1≤i<j≤τ(n)1dj−di≪1+∑1≤i<τ(n)1di+1−di\sum_{1 \le i < j \le \tau(n)} \frac{1}{d_j - d_i} \ll 1 + \sum_{1 \le i < \tau(n)} \frac{1}{d_{i + 1} - d_i} for n→∞‘n \to \infty`, i.e. Unparsed notation: \sum_{1 \le i < j \le \tau(n)} \frac{1}{d_j - d_i} \in O \left( 1 + \sum_{1 \le i < \tau(n)} \frac{1}{d_{i + 1} - d_i}) Unparsed notation: \right)?Source says disproved (Lean)Formalized
#885Is it true that, for every k≥1k \geq 1, there exist integers N1<⋯<NkN_1 < \dots < N_k such that ∣∩iD(Ni)∣≥k|\cap_i D(N_i)| \geq k?Source says openFormalized
#886Let ϵ>0\epsilon>0. Is it true that, for all large nn, the number of divisors of nn in (n1/2,n1/2+n1/2−ϵ)(n^{1/2},n^{1/2}+n^{1/2-\epsilon}) is Oϵ(1)O_\epsilon(1)?Source says openFormalized
#887Is there an absolute constant KK such that, for every C>0C > 0, if nn is sufficiently large then nn has at most KK divisors in (n12,n12+Cn14)(n^{\frac{1}{2}}, n^{\frac{1}{2}} + C n^{\frac{1}{4}}).Source says openFormalized
#893Does the limit lim⁡n→∞f(2n)f(n)\lim_{n\to\infty} \frac{f(2n)}{f(n)} tend to infinity?Source says openFormalized
#945Is it true that F(x)≤(log⁡x)O(1)F(x) \leq (\log x)^{O(1)}?Source says openFormalized
#946There are infinitely many nn such that τ(n)=τ(n+1)τ(n) = τ(n+1). Proved in [He84]. Here τ is the divisor counting function, which is σ 0 in mathlib.Source says provedFormalized
#964No statement retained — open to read what the source holdsSource says proved (Lean)No formal declaration
#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?Source says openFormalized
#1054Let f(n)f(n) be the minimal integer mm such that nn is the sum of the kk smallest divisors of mm for some k≥1k\geq 1. Show that ff is undefined at n=2n=2, i.e. we get the junk value 00.Source says openFormalized
#1099No statement retained — open to read what the source holdsSource says provedNo formal declaration
#1100No statement retained — open to read what the source holdsSource says openNo formal declaration
#1217No statement retained — open to read what the source holdsSource says provedNo formal declaration