Partitio Numerorum: sums of a prime and a number of $k$-th powers

TL;DR

This paper establishes an asymptotic lower bound on the number of representations of large integers as a sum of one prime and multiple positive integral k-th powers, extending understanding of additive number theory.

Contribution

It introduces a new asymptotic lower bound for representations involving a prime and k-th powers when the number of powers exceeds a specific threshold.

Findings

Established lower bounds for representations with prime and k-th powers

Identified threshold s ≥ c k + 4 for asymptotic estimates

Extended classical results in additive number theory

Abstract

Let $k$ be a natural number and let $c=2.134693\ldots$ be the unique real solution of the equation $2c=2+\log (5c-1)$ in $[1,\infty)$. Then, when $s\ge ck+4$, we establish an asymptotic lower bound of the expected order of magnitude for the number of representations of a large positive integer as the sum of one prime and $s$ positive integral $k$-th powers.