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Abstract

Let $\mathcal P$ denote the set of all primes, and let $\delta _{P}$ denote the relative lower density of a subset $P$ of $\mathcal {P}$. Suppose that $P_1$ and $P_2$ are two subsets of $\mathcal {P}$ satisfying $\delta _{P_1} \gt 0$ and $\delta _{P_2} \gt \sqrt{2}/2$. Then for every sufficiently large integer $n \equiv 5 \pmod{24}$, there exist $p_1 \in P_1$, $p_2 \in P_2$, and $p_3, p_4, p_5 \in \mathcal {P}$ such that $n = p_1^2 + p_2^2 + p_3^2 + p_4^2 + p_5^2.$ Moreover, we also provide two more density versions of Hua’s theorem and a square-free version of the Cauchy–Davenport type inequality.