Waring-Goldbach problem: two squares and higher powers
Journal de Théorie des Nombres de Bordeaux, Tome 28 (2016) no. 3, pp. 791-810.

Soit P r l’ensemble des nombres presque-premiers avec au plus r facteurs premiers comptés avec avec multiplicité. Dans cet article, on motre que pour tout entier pair N suffisamment grand, l’équation

N=x2+p2+p15+p25+p35+p45+p55+p65

a des solutions avec x un P 6 et les autres étant des nombres premiers. Ceci est une amélioration de résultats antérieurs de C. Hooley.

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=x2+p2+p15+p25+p35+p45+p55+p65

is solvable with x being a P 6 and the other variables primes. This result constitutes an enhancement upon that of C. Hooley.

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DOI : https://doi.org/10.5802/jtnb.964
Classification : 11P32,  11N36
Mots clés : Waring-Goldbach problem, Hardy-Littlewood method, sieve theory, almost-prime.
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Yingchun Cai. Waring-Goldbach problem: two squares and higher powers. Journal de Théorie des Nombres de Bordeaux, Tome 28 (2016) no. 3, pp. 791-810. doi : 10.5802/jtnb.964. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.964/

[1] J. Brüdern, « Sums of squares and higher powers. I », J. London Math. Soc. (2) 35 (1987), no. 2, p. 233-250. | Article | MR 1575093 | Zbl 0589.10048

[2] —, Sieves, the circle method, and waring’s problem for cubes, Mathematica Gottingensis, vol. 51, Sonderforschungsbereich Geometrie und Analysis, 1991.

[3] —, « A sieve approach to the Waring-Goldbach problem. I. Sums of four cubes », Ann. Sci. École Norm. Sup. (4) 28 (1995), no. 4, p. 461-476. | Article | MR 1334609 | Zbl 0839.11045

[4] J. Brüdern & K. Kawada, « Ternary problems in additive prime number theory », in Analytic number theory (Beijing/Kyoto, 1999), Dev. Math., vol. 6, Kluwer Acad. Publ., Dordrecht, 2002, p. 39-91. | Article | Zbl 1028.11062

[5] P. X. Gallagher, « A large sieve density estimate near σ=1 », Invent. Math. 11 (1970), p. 329-339. | Article | MR 279049 | Zbl 0219.10048

[6] H. Halberstam & H.-E. Richert, Sieve methods, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York, 1974, London Mathematical Society Monographs, No. 4, xiv+364 pp. (loose errata) pages. | Zbl 0298.10026

[7] C. Hooley, « On a new approach to various problems of Waring’s type », in Recent progress in analytic number theory, Vol. 1 (Durham, 1979), Academic Press, London-New York, 1981, p. 127-191.

[8] L. K. Hua, Additive theory of prime numbers, Translations of Mathematical Monographs, Vol. 13, American Mathematical Society, Providence, R.I., 1965, xiii+190 pages. | Article | Zbl 0192.39304

[9] H. Iwaniec, « A new form of the error term in the linear sieve », Acta Arith. 37 (1980), p. 307-320. | Article | MR 598883 | Zbl 0444.10038

[10] K. Kawada & T. D. Wooley, « On the Waring-Goldbach problem for fourth and fifth powers », Proc. London Math. Soc. (3) 83 (2001), no. 1, p. 1-50. | Article | MR 1829558 | Zbl 1016.11046

[11] K. Thanigasalam, « On admissible exponents for kth powers », Bull. Calcutta Math. Soc. 86 (1994), no. 2, p. 175-178. | Zbl 0812.11055

[12] E. C. Titchmarsh, The theory of the Riemann zeta-function, second ed., The Clarendon Press, Oxford University Press, New York, 1986, Edited and with a preface by D. R. Heath-Brown, x+412 pages. | Zbl 0601.10026

[13] R. C. Vaughan, « Sums of three cubes », Bull. London Math. Soc. 17 (1985), no. 1, p. 17-20. | Article | MR 766440 | Zbl 0562.10022

[14] —, The Hardy-Littlewood method, second ed., Cambridge Tracts in Mathematics, vol. 125, Cambridge University Press, Cambridge, 1997, xiv+232 pages. | Zbl 0868.11046

[15] I. M. Vinogradov, Elements of number theory, Dover Publications, Inc., New York, 1954, Translated by S. Kravetz, viii+227 pages. | Zbl 0057.28201

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