On monochromatic representation of sums of squares of primes

TL;DR

This paper investigates the monochromatic representation of sums of prime squares, establishing an upper bound on the number of prime squares needed to represent large integers within a fixed colour class, improving previous bounds.

Contribution

It provides a new upper bound on the minimal number of same-colour prime squares needed to represent large integers, advancing understanding of monochromatic sum representations.

Findings

Established an upper bound: s(K)  K \, ext{exp}((3\, ext{log}\, 2 + o(1)) \, ext{log}\, K / ext{log}\, ext{log}\, K)

Improved previous bound s(K)  K^{2 + \u03b5}

Results hold for K  2 and above.

Abstract

When the sequences of squares of primes is coloured with $K$ colours, where $K \geq 1$ is an integer, let $s(K)$ be the smallest integer such that each sufficiently large integer can be written as a sum of no more than $s(K)$ squares of primes, all of the same colour. We show that $s(K) \ll K \exp\left(\frac{(3\log 2 + {\rm o}(1))\log K}{\log \log K}\right)$ for $K \geq 2$. This improves on $s(K) \ll_{\epsilon} K^{2 +\epsilon}$, which is the best available upper bound for $s(K)$.