An Anti-Waring Theorem

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 \(\{n,n + 1,n + 2,\ldots\}\). 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 \( \{m^k \mid m \in \mathbb{N}, m \geq n\} \). The aim of this paper is to prove both this statement and the anti-Waring conjecture to be true.