Dominating common factor metric on rational numbers

Abstract

This study introduces a novel metric on the set of positive rational numbers, derived by pulling back the symmetric difference metric on finite sets along a generalized divisor function. We first generalize the classical notion of divisibility to rational numbers via a prime factorization- based criterion. Using this definition, each rational number is associated with a finite set of its divisors, analogous to the classical set of divisors for positive integers. Subsequently, a distance function on positive rationals is defined by d(r, s) = σ₀(r) + σ₀(s) − 2σ₀(dcf(r, s)), where σ₀(r) denotes the number of divisors of r and dcf(r, s) is a generalized greatest commondivisor. The study establishes a characterization of minimal distances and nearest neighboursin the induced metric space, reflecting the arithmetic-geometric structure of numbers in it. It was proved that, given a positive rational number, its nearest neighbours are precisely those obtained by increasing or decreasing the exponent of one of its most frequent prime factors by one. A complete description of unit spheres was provided: two positive rational numbers are at unit distance if and only if they are of the form pʳ and pʳ⁺¹ for some r ∈ Z and some p ∈ P. These results contribute to a broader understanding of how multiplicative arithmetic properties influence interactions between rational numbers and open avenues for applications of metric space theory in multiplicative number theory.