On the Largest Integer that is not a Sum of Distinct Positive $n$th Powers

Abstract

It is known that for an arbitrary positive integer \(n\) the sequence \(S(x^n)=(1^n, 2^n, \ldots)\) is complete, meaning that every sufficiently large integer is a sum of distinct \(n\)th powers of positive integers. We prove that every integer \(m\geq (b-1)2^{n-1}(r+\frac{2}{3}(b-1)(2^{2n}-1)+2(b-2))^n-2a+ab\), where \(a=n!2^{n^2}\), \(b=2^{n^3}a^{n-1}\), \(r=2^{n^2-n}a\), is a sum of distinct \(n\)th powers of positive integers.