대한수학회지 (Journal of the Korean Mathematical Society)

  • 제59권2호
  • /
  • Pages.407-420
  • /
  • 2022
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

GOLDBACH-LINNIK TYPE PROBLEMS WITH UNEQUAL POWERS OF PRIMES

  • Zhu, Li (School of Statistics and Mathematics Shanghai Lixin University of Accounting and Finance)
  • 투고 : 2021.06.01
  • 심사 : 2021.12.06
  • 발행 : 2022.03.01
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

It is proved that every sufficiently large even integer can be represented as a sum of two squares of primes, two cubes of primes, two fourth powers of primes and 17 powers of 2.

키워드

과제정보

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.

참고문헌

  1. P. X. Gallagher, Primes and powers of 2, Invent. Math. 29 (1975), no. 2, 125-142. https://doi.org/10.1007/BF01390190
  2. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, sixth edition, Oxford University Press, Oxford, 2008.
  3. 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
  4. 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
  5. 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
  6. 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
  7. Yu. V. Linnik, Prime numbers and powers of two, Trudy Nat. Inst. Steklov. Izdat. Akad. Nauk SSSR, Moscow.38 (1951), 152-169.
  8. Yu. V. Linnik, Addition of prime numbers with powers of one and the same number, Mat. Sbornik N.S. 32(74) (1953), 3-60.
  9. 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
  10. 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
  11. 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
  12. 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
  13. 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
  14. 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
  15. 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
  16. 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
  17. T. Wang, On Linnik's almost Goldbach theorem, Sci. China Ser. A 42 (1999), no. 11, 1155-1172. https://doi.org/10.1007/BF02875983
  18. 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
  19. X. D. Zhao, Goldbach-Linnik type problems on cubes of primes, Ramanujan J. https://doi.org/10.1007/s11139-020-00303-9