Abstract
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.
Similar content being viewed by others
References
Bourgain, J., Demeter, C., Guth, L.: Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three. Ann. Math. 184(2), 633–682 (2016)
Brüdern, J., Cook, R.: On simultaneous diagonal equations and inequalities. Acta Arith. 62(2), 125–149 (1992)
Cook, R.J.: Simultaneous quadratic equations. J. Lond. Math. Soc. 2(4), 319–326 (1971)
Hua, L.K.: Additive Theory of Prime Numbers. American Mathematical Society, Providence (1965)
Rogovskaya, N.N.: An Asymptotic Formula for the Number of Solutions of a System of Equations. Diophantine Approximations, Part II (Russian), 78–84, Moskov. Gos. Univ., Moscow (1986)
Schmidt, W.M.: Equations Over Finite Fields. An Elementary Approach. Lecture Notes in Mathematics, vol. 536. Springer, Berlin (1976)
Vaughan, R.C.: The Hardy–Littlewood Method, 2nd edn. Cambridge University Press, Cambridge (1997)
Vaughan, R.C.: Sommes trigonométriques sur les nombres premiers. (French). C. R. Acad. Sci. Paris Sér. A-B 285(16), A981–A983 (1976)
Wooley, T.D.: The cubic case of the main conjecture in Vinogradov’s mean value theorem. Adv. Math. 294, 532–561 (2016)
Wooley, T.D.: On simultaneous additive equations, I. Proc. Lond. Math. Soc. (3) 63(1), 1–34 (1991)
Wooley, T.D.: On simultaneous additive equations, II. J. Reine Angew. Math. 419, 141–198 (1991)
Wooley, T.D.: Rational solutions of pairs of diagonal equations, one cubic and one quadratic. Proc. Lond. Math. Soc. 3(110), 325–35 (2015)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix 1: Sage code
Appendix 1: Sage code
Code: (SageMath 8.6)
Output:
Rights and permissions
About this article
Cite this article
Talmage, A. Simultaneous cubic and quadratic diagonal equations in 12 prime variables. Ramanujan J 57, 863–905 (2022). https://doi.org/10.1007/s11139-021-00386-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-021-00386-y