Representing positive integers as a sum of a squarefree number and a small prime

TL;DR

This paper proves that almost all positive integers can be expressed as the sum of a squarefree number and a small prime, with only a few exceptions, advancing understanding of additive number theory.

Contribution

It establishes a new representation theorem for positive integers involving squarefree numbers and small primes, identifying specific exceptions.

Findings

Almost all positive integers are representable as a squarefree number plus a small prime.

Identifies a finite set of exceptions that cannot be represented in this form.

Provides a new perspective on additive decompositions involving squarefree numbers and primes.

Abstract

We prove that every positive integer $n$ which is not equal to $1$, $2$, $3$, $6$, $11$, $30$, $155$, or $247$ can be represented as a sum of a squarefree number and a prime not exceeding $\sqrt{n}$.