On an equation by primes with one Linnik prime

TL;DR

This paper proves that sufficiently large integers can be expressed as sums of three fractional powers of primes, with one prime constrained to a specific form involving a Linnik prime, for a certain range of c.

Contribution

It establishes a new representation theorem for integers as sums of prime powers with a prime of a special form, extending previous additive prime results.

Findings

Every large integer can be represented as a sum of three prime powers for 1<c<16559/15276.

One prime in the sum has the form x^2 + y^2 + 1, involving a Linnik prime.

The result extends additive number theory with prime constraints.

Abstract

Let $[\, \cdot\,]$ be the floor function. In this paper, we prove that when $1<c<\frac{16559}{15276}$, then every sufficiently large positive integer $N$ can be represented in the form \begin{equation*} N=[p^c_1]+[p^c_2]+[p^c_3]\,, \end{equation*} where $p_1,p_2,p_3$ are primes, such that $p_1=x^2 + y^2 +1$.