[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/BKMS.b170937

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

Liu, Yuhui (School of Mathematical Sciences Tongji University)

Publication Information

Bulletin of the Korean Mathematical Society / v.55, no.6, 2018 , pp. 1659-1666 More about this Journal

Abstract

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].

Keywords

Waring-Goldbach problem; Hardy-Littlewood method; almostprime; sieve theory;

Citations & Related Records

Reference

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. DOI
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. DOI
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. DOI
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.