An Anti-Waring Theorem

Abstract

It is proven that for all positive integers $k$, $n$, and $r$, every sufficiently large positive integer is the sum of $r$ or more $k$th powers of distinct elements of $\{n,n+1,n+2,\dots \}$. The case $n=1$ is the conjecture in the title of [1].

In 1770, Waring conjectured that for each positive integer $k$ there exists a $g(k)$ such that every positive integer is a sum of $g(k)$ or fewer $k$th powers of positive integers. Hilbert proved this theorem in 1909, giving rise to Waring’s problem, which asks, for each $k$, what is the smallest $g(k)$ such that the statement holds. For further details, see [3].

As a natural question arising from this problem, Johnson and Laughlin [1] proposed what they called an anti-Waring conjecture, which is the following: If $k$ and $r$ are positive integers, then every sufficiently large positive integer is the sum of $r$ or more distinct $k$th powers of positive integers. When this holds for a pair $k,r$, let $N(k,r)$ denote the smallest positive integer such that each integer $n$ greater than or equal to $N(k,r)$ is the sum of $r$ or more $k$th powers of distinct positive integers. As noted in [1], it is easy to see that, for all $r$, $N(1,r)=1+2+\cdots +r=\frac{r(r+1)}{2}$. It is also shown in [1] that $N(2,1)=N(2,2)=N(2,3)=129$.

Johnson and Laughlin further posed the question of whether given any positive integers $k$, $n$, $r$, there exists an integer $N(k,n,r)$ such that every integer $z$ greater than or equal to $N(k,n,r)$ can be written as a sum of $r$ or more distinct elements from the set $\{m^k\mid m\in \mathbb{N},m\ge n\}$. The aim of this paper is to prove both this statement and the anti-Waring conjecture to be true.