A variant of Waring's Problem for the ring of integers modulo n

2016/09/07 by David Covert, Alex Iosevich, Covert, David +3
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1609.02090

openalex publication_date 2016/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study a variant of Waring's problem for ℤ_n, the ring of integers modulo n: For a fixed integer k ≥ 2, what is the minimum number m of kth powers necessary such that x ≡ x₁^k + … + x_m^k \pmodn has a solution for every x ∈ ℤ_n? Using only elementary methods, we answer fully this question for exponents k ≤ 10, and we further discuss some intermediary cases such as categorizing the values of n such that every element in ℤ_n can be written as a sum of three squares. Hensel's Theorem for p-adic integers plays a key role. Finally, we give an application of this problem to the Erd\H os-Falconer distance problem for rings ℤ_n^d.

Related

• Waring's Problem for Polynomial Rings and the Digit Sum of Exponents
• Geometric configurations in the ring of integers modulo p^ℓ
• Representing Integers as the Sum of Two Squares in the Ring \Z_n
• Waring Numbers of Ramified p-adic Rings
• Integers that are not the sum of positive powers