Simultaneous Cubic and Quadratic Diagonal Equations In 12 Prime Variables

TL;DR

This paper proves the existence of prime solutions for a system of cubic and quadratic diagonal equations in at least 12 variables using the Hardy-Littlewood circle method, building on recent advances in number theory.

Contribution

It establishes prime solutions for the system in 12 variables and provides sufficient local solvability conditions, extending previous integer results to prime solutions.

Findings

Prime solutions exist for s ≥ 12 variables.

Conditions for local solvability modulo primes are identified.

The method builds on Wooley's work and Vinogradov's mean value theorem.

Abstract

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 $s \geq 12$, 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 $p$.