On the sum of integers from some multiplicative sets and some powers of integers

TL;DR

This paper investigates the representation of integers as sums involving norms of ideals in cyclotomic integer rings and powers of primes, establishing conditions for the infinitude of integers that cannot be so expressed.

Contribution

It proves that if certain integers cannot be expressed as sums of ideal norms and limited prime powers, then infinitely many such integers exist, extending understanding of additive properties in cyclotomic fields.

Findings

Existence of infinitely many integers not representable as sums of ideal norms and limited prime powers.

Conditions under which integers fail to be expressed as such sums for specific cyclotomic fields.

Generalization to primes p ≥ 3 and related additive number theory results.

Abstract

We show that if there exists an integer subject to some congruence conditions that cannot be written as the sum of the norm of an ideal in $\mathbb{Z}[\exp(2\pi i/2^k)]$ and at most $k$ powers of $2$, $k\geq 3$, then there are infinitely many such integers. Also, if there exists an integer that cannot be written as the sum of an integer which is the norm of an ideal in in $\mathbb{Z}[\exp(2\pi i/p)]$ and at most $p-2$ powers of $p$, where $p\geq 3$ is a prime, then there are infinitely many such integers. Finally it is shown that there are infinitely many integers not the sum of the norm of an ideal in $\mathbb{Z}[\exp(2\pi i/p)]$ and at most $p-2$ powers of $p$, for $p\geq 3$ prime.