On monochromatic representation of sums of squares of primes

Abstract

When the sequences of squares of primes is coloured with $K$ colours, where $K\ge 1$ 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 $s(K)\ll K\exp \left(\frac{(3\log 2+\mathrm{o}(1))\log K}{\log \log K}\right)$ for $K\ge 2$. This improves on $s(K){\ll }_{\epsilon }{K}^{2+\epsilon }$, which is the best available upper bound for $s(K)$.

22 pages