arXiv subjects
Publications and source records attributed to Wouter van Doorn.
For a positive integer $a$, let $b(a)$ be the smallest integer $b > a$ such that the denominator of $\frac{1}{a} + \frac{1}{a+1} + \cdots + \frac{1}{b}$ is smaller than the denominator of $\frac{1}{a} + \frac{1}{a+1} + \cdots + \frac{1}{b-1}$. Recently it was shown that the limit inferior $$\liminf_{a \to \infty} \left(\frac{b(a) - a}{\log a}\right)$$ exists and is positive. Here we find its exact value.
The Kourovka Notebook is a long-running collection of open problems in group theory. In this paper we present solutions to eight of its problems. We construct a group with exactly two maximal locally soluble normal subgroups and show that, for every $1 \le k\le n!$, there is a group containing $n$ distinct elements whose $n!$ ordered products take exactly $k$ distinct values. We also give examples showing that group order together with the statistic $\sum_g\varphi(\lvert g\rvert)$ does not determine simplicity, and we construct a surjective non-injective Rota-Baxter operator on a non-abelian group. Further, we determine the group generated by the class transpositions of moduli at most $k$, prove that every power graph of a finite group that is a cograph is chordal, show that the right-relatively convex subgroups of a right-orderable group need not form a sublattice of its subgroup lattice, and disprove a proposed rank inequality for certain $p$-group extensions. All of these solutions were autonomously discovered and formally verified in Lean by Aristotle, a formal reasoning agent developed by Harmonic.
Let $n_k$ denote the least integer $n>2k$ such that $(n-k)(n-k+1)\cdots(n-1)$ is not divisible by any prime in the interval $(k,2k)$. Confirming a conjecture of Erd\H{o}s, we prove that, for all sufficiently large $k$, $$ n_k > e^{\frac{\log^2 k}{20 \log \log k}}. $$
With $\rho = \frac{13-\sqrt{69}}{10} \approx 0.47$, it was recently established that there exist multiplicative Sidon sets (sets without any non-trivial solutions to $ab = cd$) in $\{1, 2, \ldots, n\}$ with maximal gap size $\ll_{\varepsilon} n^{\rho + \varepsilon}$. Here we improve upon this result and show that one can take $\rho = \frac{10}{33} \approx 0.303$ instead.
We show that infinitely many three-term arithmetic progressions $N, N+d, N+2d$ of powerful numbers exist with $d = 2\sqrt{N} + 1$. We further conjecture that infinitely many of these progressions consist of three consecutive terms in the sequence of powerful numbers, which would answer a question of Erd\H{o}s in the negative.
For a positive integer $n$, let $g(n)$ denote the infimum of all real numbers $L$ such that there exists a multiplicative Sidon set $A\subseteq\{1,2,\dots,n\}$ that intersects every interval $[x,x+L]\subseteq[1,n]$. S\'ark\"ozy asked for estimates on $g(n)$, and he in particular asked whether one has $g(n)\le\sqrt n$ for every $n\in\mathbb{N}$. We first show that this estimate does indeed hold, with a proof that was autonomously discovered and formally verified in Lean by Aristotle. Next, we improve the upper bound further and, with $\rho = \frac{13-\sqrt{69}}{10} < 0.47$, prove that $g(n)\ll_{\varepsilon} n^{\rho+\varepsilon}$ for every $\varepsilon > 0$.
Revisiting a $50$-year-old estimate of Choi, Erd\H{o}s and Szemer\'edi, we show that if $A \subseteq \{1, 2, \ldots, 2n\}$ satisfies $|A| \ge n + 1.2 \cdot 10^8$, then there exist five distinct integers whose pairwise sums are all contained in $A$. In order to guarantee pairwise sums of three or four integers instead, we show that one can replace the constant $1.2 \cdot 10^8$ by $1$ or $3$ respectively, which are both optimal.
For a family $(A_q)_{q\in Q}$ of subsets of a semigroup, the product intersection set records those exponents $h \in \mathbb{N}$ for which the $h$-fold product set of the intersection, $(\bigcap_q A_q)^h$, is equal to $\bigcap_q A_q^h$, the intersection of the product sets. Nathanson recently asked which subsets of $\mathbb{N}$ can occur as a product intersection set, both for arbitrary and for decreasing families $(A_q)_{q\in Q}$. We solve both problems by giving a complete classification. In particular, when $|Q| \ge 2$, we show that in either case any subset $X \subseteq \mathbb{N}$ with $1 \in X$ occurs as a product intersection set. Both classifications were autonomously discovered and formally verified in Lean by Aristotle, a formal reasoning agent developed by Harmonic.
Let $f(m)$ be the largest integer such that for every set $A = \{a_1 < \cdots < a_m\}$ of $m$ positive integers and every open interval $I$ of length $2a_m$, there exist at least $f(m)$ disjoint pairs $(a, b)$ with $a \in A$ dividing $b \in I$. Solving a problem of Erd\H{o}s, we determine $f(m)$ exactly, and show $$ f(m)=\min\bigl(m,\lceil 2\sqrt{m}\,\rceil\bigr) $$ for all $m$. The proof was obtained through an AI-assisted workflow: the proof strategy was first proposed by ChatGPT, and the detailed argument was subsequently made fully rigorous and formally verified in Lean by Aristotle. The exposition and final proofs presented here are entirely human-written. [This paper solves Problem #650 on Bloom's website "Erd\H{o}s problems".]
For fixed positive reals $t$ and $\alpha$, consider the sequence $S_t(\alpha) = (s_1, s_2, \ldots, )$ with $s_n = \left \lfloor t\alpha^n \right \rfloor$. In 1964, Graham managed to characterize those pairs $(t, \alpha)$ with $0 < t < 1$ and $1 < \alpha < 2$ for which every large enough integer can be written as the sum of distinct elements of $S_t(\alpha)$. We show that his methods can be applied to deal with many other pairs of $(t, \alpha)$ as well.
Confirming a conjecture by Erd\H os and Pomerance, we prove that there exist intervals of length $\frac{cn\log n}{\log \log n}$ that do not contain distinct multiples of $1, 2, \ldots, n$.
Let $v(k)$ be the smallest integer larger than $1$ that does not occur among the denominators in any identity of the form $$ 1=\frac1{n_1}+\cdots+\frac1{n_k}, $$ where $1 \le n_1<\cdots 0$ such that the inequality $$ v(k)\ge e^{c k^2} $$ holds for all positive integers $k$.
We answer several questions of Erd\H{o}s regarding sequences of natural numbers $A$ whose translates $n+A$ intersect with the squarefree numbers in various specified ways. For instance, we show that if every translate only contains finitely many squarefree numbers, then $A$ has zero density, although the decay rate of this density can be arbitrarily slow. On the other hand, there exist sequences $A$ with optimal density $6/\pi^2$ for which infinitely many $n$ exist such that $n+a$ is squarefree for all $a \in A$ with $a < n$. In fact, infinitely many such $n$ exist for every exponentially increasing sequence, as long as the sequence avoids at least one residue class modulo $p^2$ for all primes $p$, a property we call admissible. If one instead requires infinitely many $n$ to exist such that $n+a$ is squarefree for all $a \in A$, then $A$ can have density arbitrarily close to, but not equal to, $6/\pi^2$. Finally, we prove bounds on the growth rate of sequences $A$ for which $a+a'$ is squarefree for all $a,a' \in A$, as well as bounds on the largest admissible subset of $\{1, 2, \ldots, N\}$.
Solving a problem by Erd\H{o}s, we prove that every positive integer $n$ can be written as a sum $$n = b_{1} + b_{2} + \ldots + b_{r}$$ of distinct $3$-smooth integers with $1 \le b_{1} < b_{2} < \ldots < b_{r} < 6b_{1}$.
Erd\H{o}s and Hall defined a pair $(m, n)$ of positive integers to be interlocking, if between any pair of consecutive divisors (both larger than $1$) of $n$ (resp. $m$) there is a divisor of $m$ (resp. $n$). A positive integer is said to be separable if it belongs to an interlocking pair. We prove that the lower density of separable powers of two is positive, as well as the lower density of powers of two which are not separable. Finally, we prove that the number of interlocking pairs whose product is equal to the product of the first primes, is finite. We hereby resolve two conjectures by Erd\H{o}s and Hall.
Disproving a conjecture of Bleicher and Erd\H{o}s, we show that there exists a lacunary sequence of positive integers such that finite sums of reciprocals of its terms attain all rational numbers from a non-empty open interval. We also study several stronger variants of their original problem: determining the value of the optimal lacunarity parameter, representing rational numbers infinitely many times, finding such lacunary sequences with arbitrarily large jumps, and relating the maximal length of a filled interval to a prescribed lacunarity parameter.
Let $\frac{a_1}{b_1}, \frac{a_2}{b_2}, \ldots$ be the Farey fractions of order $n$. We then prove that the inequality $(a_l - a_k)(b_l - b_k) \ge 0$ holds for all $k$ and $l > k$ with $l-k \le \left(\frac{1}{12} - o(1) \right)n$, sharpening an old result by Erd\H{o}s. On the other hand, we will show that for all $n \ge 4$ there are $k, l$ with $k < l < k + \frac{n}{4} + 5$ for which the product $(a_l - a_k)(b_l - b_k)$ is negative.
In $1963$ Graham proved that every positive integer $n \ge 78$ can be written as a sum of distinct positive integers $a_1, a_2, \ldots, a_r$ for which $\frac{1}{a_1} + \frac{1}{a_2} + \ldots + \frac{1}{a_r}$ is equal to $1$. In the same paper he managed to further generalize this, and showed that for all positive rationals $\alpha$ and all positive integers $m$, there exists an $n_{\alpha, m}$ such that every positive integer $n \ge n_{\alpha, m}$ has a partition with distinct parts, all larger than or equal to $m$, and such that the sum of reciprocals is equal to $\alpha$. No attempt was made to estimate the quantity $n_{\alpha, m}$, however. With $n_{\alpha} := n_{\alpha, 1}$, in this paper we provide near-optimal upper bounds on $n_{\alpha}$ and $n_{\alpha, m}$, as well as bounds on the cardinality of the set $\{\alpha : n_{\alpha} \le n\}$.