The Easier Waring Problem

Abstract

Wright [1934] stated, and probably misnamed, the following variation of the well-known Waring problem concerning writing integers as sums of kth powers. The problem is to find the least n such that for all m there are natural numbers [α ₁,…,α_n] with

$$ \pm \alpha_1^k\pm \cdots \pm \alpha_1^k = m $$

for some choice of signs. We denote the least such n by v(k). Recall that the usual Waring problem requires all positive signs. For arbitrary k the best known bounds for v(k) derive from the bounds for the usual Waring problem. This gives the bound v(k) ≪ klog(k) (though it is believed that the “right” bound in both the usual Waring problem and the easier Waring problem is 0(k)). So to date, the “easier” Waring problem is not easier than the Waring problem. However, the best bounds for small k are derived in an elementary manner from solutions to the Prouhet-Tarry-Escott problem. This is discussed later in this chapter.