For a given number $N$, we consider the problem of computing two integers $1\leq r,f<N$ such that the set $$\mathcal{X}(N,r,f) = \{(a+b)-(f+\frac{Nr+1}{f}): ab=Nr\}$$ consists only of positive integers. Computing a solution to the problem is equivalent to finding a pair $(r,f)$ satisfying $l(Nr)<f \leq l(Nr+1)$, where $l(x)$ is the largest divisor of $x$ bounded by $\sqrt{x}$. This requires factoring both $Nr$ and $Nr+1$. We present a simple randomized algorithm that - avoiding factoring - computes pairs $(r,f)$. We give an exact formula for the total number of possible pairs $(r,f)$, and with the aid of empirical data we estimate that the ratio $$\frac{\phi(N)-2}{|\mathfrak{F}(N)|}$$ is approximately about $c*\log \log N$. Here, $\mathfrak{F}(N)$ is the set of unique $r$ appearing among all possible pairs $(r,f)$, $\phi(.)$ is the Euler's Totient function, and $c$ is a constant equal to 2 for prime $N$ and oscillates much for composite $N$. As a separate and independent case, we study the same problem of computing $(r,f)$ with $r>N$. We present a procedure to find such an $r$, which requires finding the least prime in an arithmetic progression.
For $2 \leq a \leq d+1$, what is the largest integer $H_{a,d} (n)$ such that every set of $n$ points in $\mathbb{R}^d$ with no $a$ points on a common $(a-2)$-flat contains a subset of $H_{a,d} (n)$ points whose determined $(a-1)$-dimensional simplices have pairwise distinct $(a-1)$-dimensional volumes? We construct $n$...
For integers $d\geq 3$, let $F_{2,d}(n)$ be the largest size of a subset of $[n]$ containing no two distinct elements whose product is a perfect $d$-th power, and let $f_{2,d}(n)$ denote the analogous quantity when the two elements need not be distinct. Fleiner, Juh\'asz, K\"ov\'er, Pach, and S\'andor proved that both...
For a family $\mathcal{F}$ of $k$-graphs, $\ex_k(n,\mathcal{F})$ denotes the maximum number of edges in an $n$-vertex $\mathcal{F}$-free $k$-graph. Let $M_{s+1}^k$ denote a matching of size $s+1$ in $k$-uniform hypergraphs. Recently, Alon and Frankl (JCTB, 2024) determined $\ex_2(n,\{M_{s+1}^2,K_{\ell+1}\})$ for all $n...
We study the reducibility over $\mathbb{Q}$ of $F_{p,e}(x)=rx^m+p^ef(x)$, where $r\in\mathbb{Z}\setminus\{0\}$, $f\in\mathbb{Z}[x]$, and $0\le m<n:=\mathrm{deg}\, f$, for primes $p$ above explicit coefficient-dependent thresholds. For arbitrary $e\ge1$, we determine the degrees, endpoint $p$-adic valuations, reductions...
For an integer $n>1$, let $P^+(n)$ be the largest prime factor of $n$, and let $P_y^+(n)$ denote the largest prime factor of $n$ not exceeding $y$. One of Erd\H{o}s and Tur\'an's conjectures asserts that the asymptotic density of integers $n$ satisfying $P^+(n)<P^+(n+1)$ is $1/2$. Furthermore, Wang conjectures that for...
For positive integers $d$ and $k$, let $n_k(d)$ be the maximum order of a graph of maximum degree at most $d$ and diameter at most $k$. We prove that $$ \lim_{d\to\infty}\frac{n_k(d)}{d^k}=1$$ for every fixed $k$, thereby resolving the asymptotic degree-diameter problem for fixed diameter and proving a conjecture of Bo...
Wouter Cames van Batenburg, Samuel Korsky· 0 citations
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