Universal sums via products of Ramanujan's theta functions

2024/10/18 by Bulkhali, Nasser Abdo Saeed, Sun, Zhi-Wei
#11D72 #11E20 #11E25 #11F27 #14H42 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2410.14605

Abstract

An integer-valued polynomial P(x,y,z) is said to be universal (over \mathbb Z) if each nonnegative integer can be written as P(x,y,z) with x,y,z∈\mathbb Z. In this paper, we mainly introduce a new technique to determine the universality of some sums in the form x(a₁x+a₂)/2+y(b₁y+b₂)/2+z(c₁z+c₂)/2 (with a₁-a₂,b₁-b₂,c₁-c₂ all even) conjectured by Sun, using various identities of Ramanujan's theta functions. For example, we prove that x(3x+1)+y(3y+2)+2z(3z+2) and x(4x+r)+y(3y+1)/2+z(7z+1)/2 (r=1,3) are universal.

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