Compositions with 3 Pairwise Coprime Parts

2020/01/31 by J. W. Thomas, Thomas, James
Engineering · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Mathematics and Applications #Number Theory (math.NT) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2001.12001

openalex publication_date 2020/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

How many ways can we write n as a sum of 3 positive integers, no pair of which share a common factor? We express this quantity in terms of the number of solutions to a certain class of linear Diophantine equations. This allows us to show that there are ∏_{p | n} ( 1- (1)/(p²) ) ∏_{q \nmid n} ( 1- (3)/(q²) ) (n²)/(2) + O(n^3/2+o(1)) such compositions, where the products are over primes that respectively do and don't divide n. This strengthens the previous result of Bubbolini, Luca, and Spiga (arXiv:1202.1670)

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