Acknowledgement
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.
References
- P. X. Gallagher, Primes and powers of 2, Invent. Math. 29 (1975), no. 2, 125-142. https://doi.org/10.1007/BF01390190
- G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, sixth edition, Oxford University Press, Oxford, 2008.
- D. R. Heath-Brown and J.-C. Puchta, Integers represented as a sum of primes and powers of two, Asian J. Math. 6 (2002), no. 3, 535-565. https://doi.org/10.4310/AJM.2002.v6.n3.a7
- K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, second edition, Graduate Texts in Mathematics, 84, Springer-Verlag, New York, 1990. https://doi.org/10.1007/978-1-4757-2103-4
- H. Li, 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 (2000), no. 3, 229-237. https://doi.org/10.4064/aa-92-3-229-237
- H. Li, 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 (2001), no. 4, 369-379. https://doi.org/10.4064/aa96-4-7
- Yu. V. Linnik, Prime numbers and powers of two, Trudy Nat. Inst. Steklov. Izdat. Akad. Nauk SSSR, Moscow.38 (1951), 152-169.
- Yu. V. Linnik, Addition of prime numbers with powers of one and the same number, Mat. Sbornik N.S. 32(74) (1953), 3-60.
- Z. Liu, Goldbach-Linnik type problems with unequal powers of primes, J. Number Theory 176 (2017), 439-448. https://doi.org/10.1016/j.jnt.2016.12.009
- J. Liu, M. Liu, and T. Wang, The number of powers of 2 in a representation of large even integers. II, Sci. China Ser. A 41 (1998), no. 12, 1255-1271. https://doi.org/10.1007/BF02882266
- Z. Liu and G. Lu, Density of two squares of primes and powers of 2, Int. J. Number Theory 7 (2011), no. 5, 1317-1329. https://doi.org/10.1142/S1793042111004605
- X. Lu, On unequal powers of primes and powers of 2, Ramanujan J. 50 (2019), no. 1, 111-121. https://doi.org/10.1007/s11139-018-0128-2
- J. Pintz and I. Z. Ruzsa, On Linnik's approximation to Goldbach's problem. I, Acta Arith. 109 (2003), no. 2, 169-194. https://doi.org/10.4064/aa109-2-6
- J. Pintz and I. Z. Ruzsa, On Linnik's approximation to Goldbach's problem. II, Acta Math. Hungar. 161 (2020), no. 2, 569-582. https://doi.org/10.1007/s10474-020-01077-8
- D. J. Platt and T. S. Trudgian, Linnik's approximation to Goldbach's conjecture, and other problems, J. Number Theory 153 (2015), 54-62. https://doi.org/10.1016/j.jnt.2015.01.008
- R. C. Vaughan, The Hardy-Littlewood Method, second edition, Cambridge Tracts in Mathematics, 125, Cambridge University Press, Cambridge, 1997. https://doi.org/10.1017/CBO9780511470929
- T. Wang, On Linnik's almost Goldbach theorem, Sci. China Ser. A 42 (1999), no. 11, 1155-1172. https://doi.org/10.1007/BF02875983
- L. Zhao, On the Waring-Goldbach problem for fourth and sixth powers, Proc. Lond. Math. Soc. (3) 108 (2014), no. 6, 1593-1622. https://doi.org/10.1112/plms/pdt072
- X. D. Zhao, Goldbach-Linnik type problems on cubes of primes, Ramanujan J. https://doi.org/10.1007/s11139-020-00303-9