Some Bounds for Number of Solutions to ax + by + cz = n and their\n Applications

2021/06/25 by Damanvir Singh Binner, Binner, Damanvir Singh
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2106.13796

openalex publication_date 2021/06/25 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

In a recent work, the present author developed an efficient method to find\nthe number of solutions of ax+by+cz=n in non-negative integer triples\n(x,y,z) where a,b,c and n are given natural numbers. In this note, we use\nthat formula to obtain some simple looking bounds for the number of solutions\nof ax+by+cz=n. Using these bounds, we solve some special cases of a problem\nrelated to the generalization of Frobenius coin problem in three variables.\nMoreover, we use these bounds to disprove a recent conjecture of He, Shiue and\nVenkat regarding the solution structure of ax+by+cz=n.\n

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