Representation of positive integers by the form x³+y³+z³-3xyz

2015/08/24 by Vladimir Shevelev, Shevelev, Vladimir
Mathematics · #11N32 #Advanced Differential Equations and Dynamical Systems #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1508.05748

openalex publication_date 2015/08/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the number ν(n) of representations of a positive integer n by the form x³+y³+z³-3xyz in the conditions 0≤ x≤ y≤ z; z≥ x+1. We proved the following results: (i) for every positive n, except for n≡±3 \pmod9, ν(n)>=1; (ii) for the exceptional n, ν(n)=0; (iii) for every prime p≠3, ν(p)=ν(2p)=1; (iv) \limsup (ν(n))=∞; (v) for every positive n, there exists k such that ν(k)=n.

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