The representation of almost all numbers as sums of unlike powers
Journal de théorie des nombres de Bordeaux, Volume 13 (2001) no. 1, p. 227-240

We prove in this article that almost all large integers have a representation as the sum of a cube, a biquadrate, ..., and a tenth power.

Nous prouvons dans cet article que presque tout entier s'écrit comme la somme d'un cube, d'un bicarré, ..., et d'une puissance dixième.

@article{JTNB_2001__13_1_227_0,
author = {Laporta, M. B. S. and Wooley, T. D.},
title = {The representation of almost all numbers as sums of unlike powers},
journal = {Journal de th\'eorie des nombres de Bordeaux},
publisher = {Universit\'e Bordeaux I},
volume = {13},
number = {1},
year = {2001},
pages = {227-240},
zbl = {1048.11074},
mrnumber = {1838083},
language = {en},
url = {http://www.numdam.org/item/JTNB_2001__13_1_227_0}
}

Laporta, M. B. S.; Wooley, T. D. The representation of almost all numbers as sums of unlike powers. Journal de théorie des nombres de Bordeaux, Volume 13 (2001) no. 1, pp. 227-240. http://www.numdam.org/item/JTNB_2001__13_1_227_0/

[1] J. Brüdern, Sums of squares and higher powers. II. J. London Math. Soc. (2) 35 (1987), 244-250. | MR 881513 | Zbl 0589.10049

[2] J. Brüdern, A problem in additive number theory. Math. Proc. Cambridge Philos. Soc. 103 (1988), 27-33. | MR 913447 | Zbl 0655.10041

[3] J. Brüdern, T.D. Wooley, On Waring's problem: two cubes and seven biquadrates. Tsukuba Math. J. 24 (2000), 387-417. | MR 1818095 | Zbl 1003.11045

[4] H. Davenport, H. Heilbronn, On Waring's problem: two cubes and one square. Proc. London Math. Soc. (2) 43 (1937), 73-104. | JFM 63.0125.02 | Zbl 0016.24601

[5] K.B. Ford, The representation of numbers as sums of unlike powers. J. London Math. Soc. (2) 51 (1995), 14-26. | MR 1310718 | Zbl 0816.11049

[6] K.B. Ford, The representation of numbers as sums of unlike powers. II. J. Amer. Math. Soc. 9 (1996), 919-940. | MR 1325794 | Zbl 0866.11054

[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 (1981), 127-191. | MR 637346 | Zbl 0463.10037

[8] K.F. Roth, Proof that almost all positive integers are sums of a square, a positive cube and a fourth power. J. London Math. Soc. 24 (1949), 4-13. | MR 28336 | Zbl 0032.01401

[9] K.F. Roth, A problem in additive number theory. Proc. London Math. Soc. (2) 53 (1951), 381-395. | MR 41874 | Zbl 0044.03601

[10] K. Thanigasalam, On additive number theory. Acta Arith. 13 (1967/68), 237-258. | MR 222044 | Zbl 0155.09002

[11] K. Thanigasalam, On sums of powers and a related problem. Acta Arith. 36 (1980), 125-141. | MR 581911 | Zbl 0354.10016

[12] K. Thanigasalam, On certain additive representations of integers. Portugal. Math. 42 (1983/84), 447-465. | MR 836123 | Zbl 0574.10048

[13] R.C. Vaughan, On the representation of numbers as sums of powers of natural numbers. Proc. London Math. Soc. (3) 21 (1970), 160-180. | MR 272734 | Zbl 0206.06103

[14] R.C. Vaughan, On sums of mixed powers. J. London Math. Soc. (2) 3 (1971), 677-688. | MR 294279 | Zbl 0221.10050

[15] R.C. Vaughan, A ternary additive problem. Proc. London Math. Soc. (3) 41 (1980), 516-532. | MR 591653 | Zbl 0446.10042

[16] R.C. Vaughan, On Waring's problem for cubes. J. Reine Angew. Math. 365 (1986), 122-170. | MR 826156 | Zbl 0574.10046

[17] R.C. Vaughan, On Waring's problem for smaller exponents. Proc. London Math. Soc. (3) 52 (1986), 445-463. | MR 833645 | Zbl 0601.10035

[18] R.C. Vaughan, A new iterative method in Waring's problem. Acta Math. 162 (1989), 1-71. | MR 981199 | Zbl 0665.10033

[19] R.C. Vaughan, The Hardy-Littlewood method. Cambridge Tract No. 125, 2nd Edition, Cambridge University Press, 1997. | MR 1435742 | Zbl 0868.11046

[20] R.C. Vaughan, T.D. Wooley, On Waring's problem: some refinements. Proc. London Math. Soc. (3) 63 (1991), 35-68. | MR 1105718 | Zbl 0736.11058

[21] R.C. Vaughan, T.D. Wooley, Further improvements in Waring's problem. Acta Math. 174 (1995), 147-240. | MR 1351319 | Zbl 0849.11075

[22] R.C. Vaughan, T.D. Wooley, Further improvements in Waring's problem, IV: higher powers. Acta Arith. 94 (2000), 203-285. | MR 1776896 | Zbl 0972.11092

[23] T.D. Wooley, On simultaneous additive equations, II. J. Reine Angew. Math. 419 (1991), 141-198. | MR 1116923 | Zbl 0721.11011

[24] T.D. Wooley, Large improvements in Waring's problem. Ann. of Math. (2) 135 (1992), 131-164. | MR 1147960 | Zbl 0754.11026

[25] T.D. Wooley, New estimates for smooth Weyl sums. J. London Math. Soc. (2) 51 (1995), 1-13. | MR 1310717 | Zbl 0833.11041

[26] T.D. Wooley, Breaking classical convexity in Waring's problem: sums of cubes and quasi-diagonal behaviour. Invent. Math. 122 (1995), 421-451. | MR 1359599 | Zbl 0851.11055