Representations by x₁²+2x₂²+x₃²+x₄²+x₁x₃+x₁x₄+x₂x₄

2011/02/28 by Ick Sun Eum, Eum, Ick Sun, Dong Hwa Shin +3
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1102.5746

openalex publication_date 2011/02/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let r_Q(n) be the representation number of a nonnegative integer n by the quaternary quadratic form Q=x₁²+2x₂²+x₃²+x₄²+x₁x₃+x₁x₄+x₂x₄. We first prove the identity r_Q(p²n)=r_Q(p²)r_Q(n)/r_Q(1) for any prime p different from 13 and any positive integer n prime to p, which was conjectured in [Eum et al, A modularity criterion for Klein forms, with an application to modular forms of level 13, J. Math. Anal. Appl. 375 (2011), 28--41]. And, we explicitly determine a concise formula for the number r_Q(n²) as well for any integer n.

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