The system of equations

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.

方程组

具有原溶液\（（P_1，\ ldots，P_S）\）为\（S \ GE 12 \） ，假设系统具有解模每个素p。这可以通过Hardy–Littlewood圆法证明，该方法建立在Wooley在相应系统上的整数上的工作以及基于Vinogradov中值定理的最新结果的基础上。另外，给出了一组局部可解性的充分条件：如果两个方程都是模2可解的，则二次方程是模3可解的，并且对于每个素数p，每个\（u_i \），\（v_i \）不是p的零模，则系统具有对每个素数p求模的解。