Simultaneous cubic and quadratic diagonal equations in 12 prime variables

Abstract

The system of equations $$\begin{aligned}&u_1p_1^2 + \cdots + u_sp_s^2 = 0,\\&v_1p_1^3 + \cdots + v_sp_s^3 = 0 \end{aligned}$$ has prime solutions $$(p_1, \ldots , p_s)$$ for $$s \ge 12$$ , assuming that the system has solutions modulo each prime p. This is proved via the Hardy–Littlewood circle method, building on Wooley’s work on the corresponding system over the integers and recent results on Vinogradov’s mean value theorem. Additionally, a set of sufficient conditions for local solvability is given: If both equations are solvable modulo 2, the quadratic equation is solvable modulo 3, and for each prime p at least 7 of each of the $$u_i$$ , $$v_i$$ are not zero modulo p, then the system has solutions modulo each prime p.