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

Abstract

We study a variant of Waring's problem for $\mathbb{Z}_n$, the ring of integers modulo $n$: For a fixed integer $k \geq 2$, what is the minimum number $m$ of $k$th powers necessary such that $x \equiv x_1^k + \dots + x_m^k \pmod{n}$ has a solution for every $x \in \mathbb{Z}_n$? Using only elementary methods, we answer fully this question for exponents $k \leq 10$, and we further discuss some intermediary cases such as categorizing the values of $n$ such that every element in $\mathbb{Z}_n$ 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 $\mathbb{Z}_n^d$.

Paper withdrawn as main results are not original as we once thought. Main results originally proved in "Some Problems of 'Partitio Numerorum' (VIII): The Number {Gamma}(k) in Waring's Problem," by G.H. Hardy and J.E. Littlewood, appearing in Proc. London Math. Soc. (2) S2-28 (1928), 518--542