Abstract
In this paper, it is proved by a different method that every sufficiently large odd integer can be written as a sum of one prime, two squares of primes and 17 powers of 2, which improves the previous result \(k=31\).
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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 \(2\). Asian J. Math. 6, 535–565 (2002)
Hua, L.K.: Some results in the additive prime number theory. Q. J. Math. (Oxford) 9, 68–80 (1938)
Kumchev, A.V.: On Weyl sums over primes and almost primes. Mich. Math. J. 54, 243–268 (2006)
Li, H.Z.: The number of powers of \(2\) in a representation of large even integersby 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 integersby sums of such powers and of two primes II. Acta Arith. 96, 369–379 (2001)
Li, H.Z.: Representation of odd integers as the sum of one prime, two squares of primes and powers of \(2\). Acta Arith. 128, 223–233 (2007)
Linnik, Y.V.: Primes numbers and powers of \(2\). Trudy Mat. Inst. Steklov. 38, 152–169 (1951). (in Russian)
Linnik, Y.V.: Addition of prime numbers with powers of one and the same number. Mat. Sb.(N. S.) 32, 3–60 (1953). (in Russian)
Liu, H.F.: Two results on powers of two in Waring–Goldbach type problems. Int. J. Number Theory 12, 1813–1825 (2016)
Liu, J.Y., Liu, M.C., Wang, T.Z.: The number of powers of \(2\) in a representation of large even integers(II). Sci. China Ser. A 41, 1255–1271 (1998)
Liu, J.Y., Liu, M.C., Zhan, T.: Squares of primes and powers of \(2\). Monatsh. Math. 128, 283–313 (1999)
Liu, J.Y., Liu, M.C.: Representation of even integers as sums of squares of primes and powers of \(2\). J. Number Theory 83, 202–225 (2000)
Liu, T.: Representation of odd integers as the sum of one prime, two squares of primes and powers of \(2\). Acta Arith. 115, 97–118 (2004)
Liu, J.Y., Lü, G.S.: Four squares of primes and \(165\) powers of \(2\). Acta Arith. 114, 55–70 (2004)
Liu, Z.X.: One prime, two squares of primes and powers of 2. Acta Math. Hung. 143, 3–12 (2014)
Liu, Z.X., Lü, G.S.: Density of two squares of primes and powers of 2. Int. J. Number Theory 7, 1317–1329 (2011)
Lü, G.S., Sun, H.W.: Integers represented as the sum of one prime, two squares of primes and powers of 2. Proc. Am. Math. Soc. 137, 1185–4191 (2009)
Platt, D.J., Trudgian, T.S.: Linnik’s approximation to Goldbach’s conjecture, and other problems. J. Number Theory 153, 54–62 (2015)
Pintz, J., Ruzsa, I.Z.: On Linnik’s approximation to Goldbach’s problem I. Acta Arith. 109, 169–194 (2003)
Pintz, J.: A note on Romanov’s constant. Acta Math. Hung. 112, 1–14 (2006)
Wang, T.Z.: On Linnik’s almost Goldbach theorem. Sci. China Ser. A 42, 1155–1172 (1999)
Wu, J.: Chen’s double sieve, Goldbach conjecture and twin prime theorem. Acta Arith. 114, 215–273 (2004)
Zhao, L.L.: Some results on Waring–Goldbach type problems, Ph.D thesis, Hong Kong Uniersity (2012)
Zhao, L.L.: On unequal powers of primes and powers of \(2\). Acta Math. Hung. 142, 405–420 (2015)
Acknowledgements
The author is grateful to the reviewers for their valuable suggestions and comments. This work is supported in part by NSFC (Nos. 11771252, 11531008), IRT16R43, and Taishan Scholars Project.
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Communicated by A. Constantin.
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Lü, G. On sum of one prime, two squares of primes and powers of 2. Monatsh Math 187, 113–123 (2018). https://doi.org/10.1007/s00605-017-1104-4
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DOI: https://doi.org/10.1007/s00605-017-1104-4