The representation of almost all numbers as sums of unlike powers
Journal de Théorie des Nombres de Bordeaux, Tome 13 (2001) no. 1, pp. 227-240.

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.

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.

@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},
     pages = {227--240},
     publisher = {Universit\'e Bordeaux I},
     volume = {13},
     number = {1},
     year = {2001},
     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, Tome 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. | EuDML 204827 | MR 222044 | Zbl 0155.09002

[11] K. Thanigasalam, On sums of powers and a related problem. Acta Arith. 36 (1980), 125-141. | EuDML 205663 | MR 581911 | Zbl 0354.10016

[12] K. Thanigasalam, On certain additive representations of integers. Portugal. Math. 42 (1983/84), 447-465. | EuDML 115560 | 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. | EuDML 152808 | 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. | EuDML 207431 | MR 1776896 | Zbl 0972.11092

[23] T.D. Wooley, On simultaneous additive equations, II. J. Reine Angew. Math. 419 (1991), 141-198. | EuDML 153348 | 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. | EuDML 144327 | MR 1359599 | Zbl 0851.11055