The exceptional set for integers of the form $[(p_1)^c] + [(p_2)^c]$

TL;DR

This paper investigates the representation of integers as sums of fractional powers of primes, establishing bounds on the count of integers that cannot be expressed in such a form for a specific range of c.

Contribution

It provides an explicit bound on the number of integers not representable as sums of fractional powers of primes for 1 < c < 24/19.

Findings

Number of non-representable integers up to N is O(N^{1-σ+ε})

Explicit function σ(c) is derived and used in bounds

Results extend understanding of additive prime number theory

Abstract

Let $1 < c < 24/19$. We show that the number of integers $n \le N$ that cannot be written as $[p_1^c] + [p_2^c]$ ($p_1$, $p_2$ primes) is $O(N^{1-\sigma+\varepsilon})$. Here $\sigma$ is a positive function of $c$ (given explicitly) and $\varepsilon$ is an arbitrary positive number.