Abstract
It is proved that every sufficiently large even integer can be represented as a sum of two squares of primes, four cubes of primes and 14 powers of 2.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Gallagher, P.X.: Primes and powers of 2. Invent. Math. 29, 125–142 (1975)
Heath-Brown, D.R., Puchta, J.C.: Integers represented as a sum of primes and powers of two. Asian J. Math. 6, 535–565 (2002)
Kawada, K., Wooley, T.D.: On the Waring-Goldbach problem for fourth and fifth powers. Proc. Lond. Math. Soc. (3) 83(1), 1–50 (2001)
Kumchev, A.V.: On Weyl sums over primes and almost primes. Michigan Math. J. 54(2), 243–268 (2006)
Li, H.Z.: The number of powers of 2 in a representation of large even integers by sums of such powers and of two primes. Acta Arith. 92, 229–237 (2000)
Li, H.Z.: The number of powers of 2 in a representation of large even integers by sums of such powers and of two primes. II. Acta Arith. 96, 369–379 (2001)
Linnik, Yu.V.: Prime numbers and powers of two. Tr. Mat. Inst. Steklova 38, 151–169 (1951). ((in Russian))
Linnik, Yu. V.: Addition of prime numbers with powers of one and the same number. Mat. Sb. (N.S.) 32, 3–60 (1953) (in Russian)
Liu, J.Y., Liu, M.C., Wang, T.Z.: The number of powers of two in a representation of large even integers. II. Sci. China Ser. A 41, 1255–1271 (1998)
Liu, Z.X., Lü, G.S.: Density of two squares of primes and powers of 2. Int. J. Number Theory 7, 1317–1329 (2011)
Liu, Z.X., Lü, G.S.: Two results on powers of 2 in Waring-Goldbach problem. J. Number Theory 131, 716–736 (2011)
Liu, Z.X.: Goldbach–Linnik type problems with unequal powers of primes. J. Number Theory 176, 439–448 (2017)
Pintz, J., Ruzsa, I.Z.: On Linnik’s approximation to Goldbach’s problem. I. Acta Arith. 109, 169–194 (2003)
Pintz, J., Ruzsa, I.Z.: On Linnik’s approximation to Goldbach’s problem. II. Acta Math. Hungar. 161, 569–582 (2020)
Platt, D.J., Trudgian, T.S.: Linnik’s approximation to Goldbach’s conjecture, and other problems. J. Number Theory 153, 54–62 (2015)
Vaughan, R.C.: The Hardy-Littlewood Method, Second edition, Cambridge Tracts in Mathematics, vol. 125. Cambridge University Press, Cambridge (1997)
Wang, T.Z.: On Linnik’s almost Goldbach theorem. Sci. China Ser. A 42, 1155–1172 (1999)
Zhao, L.L.: On the Waring-Goldbach problem for fourth and sixth powers. Proc. Lond. Math. Soc. (3) 108, 1593–1622 (2014)
Zhao, L.L.: On unequal powers of primes and powers of 2. Acta Math. Hungar. 146, 405–420 (2015)
Zhao, X.D.: Two prime squares, four prime cubes and powers of 2. Acta Arith. 187, 143–150 (2019)
Acknowledgements
We thank the referees for their time and comments. The author would like to express the most sincere gratitude to Professor Yingchun Cai for his valuable advice and constant encouragement.
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 was supported by The National Natural Science Foundation of China (Grant No. 11771333).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) 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
Zhu, L. Two prime squares, four prime cubes and powers of 2. Ramanujan J 60, 427–445 (2023). https://doi.org/10.1007/s11139-022-00671-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-022-00671-4