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On representation of integers by sums of a cube and three cubes of primes. (English) Zbl 1101.11044

Let \(N\) be a large positive integer and let \(E(N)\) denote the number of positive integers \(n\leq N\) that cannot be written in the form \[ n=m^3+p_2^3+p_3^3+p_4^3 \] with positive integer \(m\) and primes \(p_j\). For any positive number \(\varepsilon\) the authors prove that \(E(N)\ll N^{17/18+\varepsilon}\).

MSC:

11P32 Goldbach-type theorems; other additive questions involving primes
11P55 Applications of the Hardy-Littlewood method
11D25 Cubic and quartic Diophantine equations
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References:

[1] J. Brüdern, A sieve approach to the Waring–Goldbach problem. I. Sums of four cubes, Ann. Sci. École Norm. Sup (4) 28 (1995), 461–476. · Zbl 0839.11045
[2] K. Kawada, Note on the sum of cubes of primes and an almost prime, Arch. Math. (Basel) 69 (1997), 13–19. · Zbl 0882.11057 · doi:10.1007/s000130050088
[3] K. Kawada and T. D. Wooley, On the Waring–Goldbach problem for fourth and fifth powers, Proc. London Math. Soc. (3) 83 (2001), 1–50. · Zbl 1016.11046 · doi:10.1112/plms/83.1.1
[4] J. Y. Liu, On Lagrange’s theorem with prime variables, Quart. J. Math. Oxford Ser. (2) 54 (2003), 453–462. · Zbl 1080.11071 · doi:10.1093/qjmath/54.4.453
[5] X. Ren, The exceptional set in Roth’s theorem concerning a cube and three cubes of primes, Quart. J. Math. Oxford Ser. (2) 52 (2001), 107–126. · Zbl 0991.11056 · doi:10.1093/qjmath/52.1.107
[6] ——, On exponential sums over primes and applications in the Waring–Goldbach problem, Sci. China Ser. A 48 (2005), 785–797. · Zbl 1100.11025 · doi:10.1360/03ys0341
[7] X. Ren and K. M. Tsang, On Roth’s theorem concerning a cube and three cubes of primes, Quart. J. Math. Oxford Ser. (2) 55 (2004), 357–374. · Zbl 1060.11060 · doi:10.1093/qjmath/55.3.357
[8] K. F. Roth, On Waring’s problem for cubes, Proc. London Math. Soc. (2) 53 (1951), 268–279. · Zbl 0043.27303 · doi:10.1112/plms/s2-53.4.268
[9] R. C. Vaughan, Sums of three cubes, Bull. London Math. Soc. 17 (1985), 17–20. · Zbl 0562.10022 · doi:10.1112/blms/17.1.17
[10] ——, The Hardy–Littlewood method, 2nd ed., Cambridge Tracts in Math., 125, Cambridge Univ. Press, 1997. · Zbl 0868.11046
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