2025/04/27 by Adenwalla, Sarosh
#11A05 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2504.19196
Ron Graham introduced a function, g(n), on the non-negative integers, in the 1986 Issue 3 Problems column of Mathematical Magazine: For each non-negative integer n, g(n) is the least integer s so that the integers n + 1, n + 2, … , s-1, s contain a subset of integers, the product of whose members with n is a square. Recently, many results about g(n) were proved in [Kagey and Rajesh, ArXiv:2410.04728, 2024] and they conjectured a characterization of which n satisfied g(n)=2n. For m≥ 2, they also introduced generalizations of g(n) to m-th powers to explore. In this paper, we prove their conjecture and provide some results about these generalisations.