Matrix Waring Problem-III

Abstract

Let $n,k$ be positive integers and $q$ be a prime power. We prove that there exists a real constant $C_k$ depending only on $k$ such that if $\#M_n({\mathbb{F}}_q)>C_k$ then every element in $M_n({\mathbb{F}}_q)$ is a sum of two $k$th powers, proving Larsen's conjecture.