DOI QR코드

DOI QR Code

ON A WARING-GOLDBACH PROBLEM INVOLVING SQUARES, CUBES AND BIQUADRATES

  • Liu, Yuhui (School of Mathematical Sciences Tongji University)
  • 투고 : 2017.10.26
  • 심사 : 2018.10.11
  • 발행 : 2018.11.30

초록

Let $P_r$ denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, it is proved that for every sufficiently large even integer N, the equation $$N=x^2+p_1^2+p_2^3+p_3^3+p_4^4+p_5^4$$ is solvable with x being an almost-prime $P_4$ and the other variables primes. This result constitutes an improvement upon that of $L{\ddot{u}}$ [7].

과제정보

연구 과제 주관 기관 : National Natural Science Foundation of China

참고문헌

  1. J. Brudern, Sieves, the circle method and Waring's problem for cubes, Mathematica Gottingenis 51 (1991).
  2. J. Brudern, A sieve approach to the Waring-Goldbach problem I. Sums of four cubes, Ann. Sci. Ec. Norm. Sup. 28 (1995), no. 4, 461-476. https://doi.org/10.24033/asens.1721
  3. J. Brudern and K. Kawada, Ternary problems in additive prime number theory, in Analytic number theory (Beijing/Kyoto, 1999), 39-91, Dev. Math., 6, Kluwer Acad. Publ., Dordrecht, 2002.
  4. Y. Cai, On Waring-Goldbach problem involving fourth powers, J. Number Theory 131 (2011), no. 8, 1347-1362. https://doi.org/10.1016/j.jnt.2010.12.007
  5. H. Halberstam and H.-E. Richert, Sieve Methods, Academic Press, London, 1974.
  6. H. Iwaniec, A new form of the error term in the linear sieve, Acta Arith. 37 (1980), 307-320. https://doi.org/10.4064/aa-37-1-307-320
  7. X. D. Lu, The Waring-Goldbach problem: two squares, two cubes and two biquadrates, Chinese J. Contemp. Math. 36 (2015), no. 2, 145-156; translated from Chinese Ann. Math. Ser. A 36 (2015), no. 2, 161-174.
  8. R. C. Vaughan, On the representation of numbers as sums of squares, cubes and fourth powers and on the representation of numbers as sums of powers of primes, University of London, 1969.
  9. R. C. Vaughan, The Hardy-Littlewood Method, second edition, Cambridge Tracts in Mathematics, 125, Cambridge University Press, Cambridge, 1997.