Abstract
In this paper, we prove that every sufficiently large even integer can be represented as a sum of two squares of primes, four cubes of primes, and 21 powers of 2.
Similar content being viewed by others
References
Hua, L.K.: Additive Theory of Prime Numbers. Science Press. Amer. Math. Soc, Providence (1965)
Kumchev, A.V.: On Weyl sums over primes and almost primes. Michigan Math. J. 54, 243–268 (2006)
Languasco, A., Zaccagnini, A.: On a Diophantine problem with two primes and \(s\) powers of two. Acta Arith. 145, 193–208 (2010)
Linnik, Y.V.: Prime numbers and powers of two. Trudy Mat. Inst. Steklov 38, 152–169 (1951)
Linnik, Y.V.: Addition of prime numbers with powers of one and the same number. Mat. Sbornik N. S. 32, 3–60 (1953)
Liu, Z.X.: Goldbach–Linnik type problems with unequal powers of primes. J. Number Theory 176, 439–448 (2017)
Liu, Z.X., Lü, G.S.: Two results on powers of 2 in Waring–Goldbach problem. J. Number Theory 131, 716–736 (2011)
Lü, X.D.: On unequal powers of primes and powers of 2. Ramanujan J. 50, 111–121 (2019)
Zhao, L.L.: On the Waring–Goldbach problem for fourth and sixth powers. Proc. Lond. Math. Soc. 108, 1593–1622 (2014)
Zhao, L.L.: The additive problem with one cube and three cubes of primes. Michigan Math. J. 63, 763–779 (2014)
Zhao, L.L.: On unequal powers of primes and powers of 2. Acta Math. Hung. 146, 405–420 (2015)
Zhao, X.D.: Two prime squares, four prime cubes and powers of 2. Acta Arith. 187, 143–150 (2019)
Zhao, X.D.: Goldbach–Linnik type problems on cubes of primes. Ramanujan J. 57, 239–251 (2022)
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.
This work is supported by the National Natural Science Foundation of China (Grant No. 11871367, Grant No. 11871307, and Grant No. 12031008) and Natural Science Foundation of Tianjin City (Grant No. 19JCQNJC14200).
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Zhang, R. Goldbach–Linnik type problems with mixed powers of primes. Ramanujan J 60, 1069–1080 (2023). https://doi.org/10.1007/s11139-022-00662-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-022-00662-5