2025/10/22 by Stijn Cambie, Wouter van Doorn, Cambie, Stijn +1
Mathematics · #11A51 #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2510.19727
openalex publication_date 2025/10/22 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
Erdős 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 Erdős and Hall.