A new variant of the Hilbert-Waring problem

Abstract:

In this paper, we propose a new variant of Waring’s problem: to express a positive integer $n$ as a sum of $s$ positive integers whose product is a $k$-th power. We define, in a similar way to that done for $g(k)$ and $G(k)$ in Waring’s problem, numbers $g’(k)$ and $G’(k)$. We obtain $g’(k)=2k-1$, $G’(p)\leq p+1$ for primes $p$, $G’(2p)\leq 2p+2$ for odd primes $p$. Moreover, we obtain several interesting results and make two conjectures about $G’(3)$ and $G’(4)$.