Abstract
For any integer K≥1, let s(K) be the smallest integer such that when the set of squares of the prime numbers is coloured in K colours, 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)≪Kexp((3log2+o(1))logKloglogK) for K≥2. This upper bound for s(K) is close to optimal and improves on s(K)≪ϵK2+ϵ, which is the best available upper bound for s(K).
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