The representation of almost all numbers as sums of unlike powers

M. B. S. Laporta; T. D. Wooley

Journal de théorie des nombres de Bordeaux (2001)

  • Volume: 13, Issue: 1, page 227-240
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Laporta, M. B. S., and Wooley, T. D.. "The representation of almost all numbers as sums of unlike powers." Journal de théorie des nombres de Bordeaux 13.1 (2001): 227-240. <http://eudml.org/doc/248701>.

@article{Laporta2001,
abstract = {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.},
author = {Laporta, M. B. S., Wooley, T. D.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {sums of unlike powers; Waring's problem; smooth Weyl sums},
language = {eng},
number = {1},
pages = {227-240},
publisher = {Université Bordeaux I},
title = {The representation of almost all numbers as sums of unlike powers},
url = {http://eudml.org/doc/248701},
volume = {13},
year = {2001},
}

TY - JOUR
AU - Laporta, M. B. S.
AU - Wooley, T. D.
TI - The representation of almost all numbers as sums of unlike powers
JO - Journal de théorie des nombres de Bordeaux
PY - 2001
PB - Université Bordeaux I
VL - 13
IS - 1
SP - 227
EP - 240
AB - 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.
LA - eng
KW - sums of unlike powers; Waring's problem; smooth Weyl sums
UR - http://eudml.org/doc/248701
ER -

References

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  1. [1] J. Brüdern, Sums of squares and higher powers. II. J. London Math. Soc. (2) 35 (1987), 244-250. Zbl0589.10049MR881513
  2. [2] J. Brüdern, A problem in additive number theory. Math. Proc. Cambridge Philos. Soc.103 (1988), 27-33. Zbl0655.10041MR913447
  3. [3] J. Brüdern, T.D. Wooley, On Waring's problem: two cubes and seven biquadrates. Tsukuba Math. J.24 (2000), 387-417. Zbl1003.11045MR1818095
  4. [4] H. Davenport, H. Heilbronn, On Waring's problem: two cubes and one square. Proc. London Math. Soc. (2) 43 (1937), 73-104. Zbl0016.24601JFM63.0125.02
  5. [5] K.B. Ford, The representation of numbers as sums of unlike powers. J. London Math. Soc. (2) 51 (1995), 14-26. Zbl0816.11049MR1310718
  6. [6] K.B. Ford, The representation of numbers as sums of unlike powers. II. J. Amer. Math. Soc.9 (1996), 919-940. Zbl0866.11054MR1325794
  7. [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. Zbl0463.10037MR637346
  8. [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. Zbl0032.01401MR28336
  9. [9] K.F. Roth, A problem in additive number theory. Proc. London Math. Soc. (2) 53 (1951), 381-395. Zbl0044.03601MR41874
  10. [10] K. Thanigasalam, On additive number theory. Acta Arith.13 (1967/68), 237-258. Zbl0155.09002MR222044
  11. [11] K. Thanigasalam, On sums of powers and a related problem. Acta Arith.36 (1980), 125-141. Zbl0354.10016MR581911
  12. [12] K. Thanigasalam, On certain additive representations of integers. Portugal. Math.42 (1983/84), 447-465. Zbl0574.10048MR836123
  13. [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. Zbl0206.06103MR272734
  14. [14] R.C. Vaughan, On sums of mixed powers. J. London Math. Soc. (2) 3 (1971), 677-688. Zbl0221.10050MR294279
  15. [15] R.C. Vaughan, A ternary additive problem. Proc. London Math. Soc. (3) 41 (1980), 516-532. Zbl0446.10042MR591653
  16. [16] R.C. Vaughan, On Waring's problem for cubes. J. Reine Angew. Math.365 (1986), 122-170. Zbl0574.10046MR826156
  17. [17] R.C. Vaughan, On Waring's problem for smaller exponents. Proc. London Math. Soc. (3) 52 (1986), 445-463. Zbl0601.10035MR833645
  18. [18] R.C. Vaughan, A new iterative method in Waring's problem. Acta Math.162 (1989), 1-71. Zbl0665.10033MR981199
  19. [19] R.C. Vaughan, The Hardy-Littlewood method. Cambridge Tract No. 125, 2nd Edition, Cambridge University Press, 1997. Zbl0868.11046MR1435742
  20. [20] R.C. Vaughan, T.D. Wooley, On Waring's problem: some refinements. Proc. London Math. Soc. (3) 63 (1991), 35-68. Zbl0736.11058MR1105718
  21. [21] R.C. Vaughan, T.D. Wooley, Further improvements in Waring's problem. Acta Math.174 (1995), 147-240. Zbl0849.11075MR1351319
  22. [22] R.C. Vaughan, T.D. Wooley, Further improvements in Waring's problem, IV: higher powers. Acta Arith.94 (2000), 203-285. Zbl0972.11092MR1776896
  23. [23] T.D. Wooley, On simultaneous additive equations, II. J. Reine Angew. Math.419 (1991), 141-198. Zbl0721.11011MR1116923
  24. [24] T.D. Wooley, Large improvements in Waring's problem. Ann. of Math. (2) 135 (1992), 131-164. Zbl0754.11026MR1147960
  25. [25] T.D. Wooley, New estimates for smooth Weyl sums. J. London Math. Soc. (2) 51 (1995), 1-13. Zbl0833.11041MR1310717
  26. [26] T.D. Wooley, Breaking classical convexity in Waring's problem: sums of cubes and quasi-diagonal behaviour. Invent. Math.122 (1995), 421-451. Zbl0851.11055MR1359599

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