Universal sums via products of Ramanujan's theta functions
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\in\mathbb{Z} $. In this paper, we mainly introduce a new technique to determine the universality of some sums in the form $ x(a_1x+a_2)/2+y(b_1y+b_2)/2+z(c_1z+c_2)/2 $ (with $ a_1-a_2, b_1-b_2, c_1-c_2 $ 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.
Fields of Study & Subjects
Advanced Mathematical Identities
Analytic Number Theory Research
Mathematical Inequalities and Applications
Publication details
- Journal / source
- Frontiers in Combinatorics and Number Theory
- Volume
- 3
- Issue
- 0
- Pages
- 68–87
- Publication date
- 1 January 2026
- Language
- EN