On an equation with prime numbers close to squares

TL;DR

This paper proves that for certain exponents, large integers can be expressed as sums of three primes near perfect squares, expanding understanding of prime representations in additive number theory.

Contribution

It establishes a new result on representing large integers as sums of three primes close to squares for specific fractional exponents.

Findings

Every sufficiently large integer can be expressed as a sum of three primes near squares for 1<c<37/36.

The representation involves primes close to squares, extending classical prime sum problems.

The result applies to a range of fractional exponents, broadening the scope of additive prime number theory.

Abstract

Let $[\, \cdot\,]$ be the floor function. In this paper, we show that when $1<c<37/36$, 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 close to squares.