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The system of equations \[ u_1p_1^2 + \ldots + u_sp_s^2 = 0 \] \[ v_1p_1^3 +\ldots + v_sp_s^3 = 0 \] has prime solutions $(p_1, \ldots, p_s)$ for $s \geq12$, assuming that the system has solutions modulo each prime $p$. This isproved via the Hardy-Littlewood circle method, building on Wooley's work on thecorresponding system over the integers and recent results on Vinogradov's meanvalue theorem. Additionally, a set of sufficient conditions for localsolvability is given: If both equations are solvable modulo 2, the quadraticequation 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 eachprime $p$.

Simultaneous Cubic and Quadratic Diagonal Equations In 12 Prime Variables