Representations of Numbers as Sums and Differences of Unlike Powers
Jabara, Enrico
Bollettino dell'Unione Matematica Italiana, Tome 3 (2010), p. 169-177 / Harvested from Biblioteca Digitale Italiana di Matematica
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In this paper we prove that every n∈𝐙 can be written as n=ϵ1⁢x12+ϵ2⁢x23+ϵ3⁢x34+ϵ4⁢x45 and as n=ϵ1⁢x13+ϵ2⁢x24+ϵ3⁢x35+ϵ4⁢x46+ϵ5⁢x57+ϵ6⁢x68+ϵ7⁢x79+ϵ8⁢x810 with xi∈𝐙 and ϵi∈{-1,1}. We also prove some other results on numbers expressible as sums or differences of unlike powers.

Publié le : 2010-02-01
@article{BUMI_2010_9_3_1_169_0,
     author = {Enrico Jabara},
     title = {Representations of Numbers as Sums and Differences of Unlike Powers},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {3},
     year = {2010},
     pages = {169-177},
     zbl = {1198.11037},
     mrnumber = {2605918},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2010_9_3_1_169_0}
}

Jabara, Enrico. Representations of Numbers as Sums and Differences of Unlike Powers. Bollettino dell'Unione Matematica Italiana, Tome 3 (2010) pp. 169-177. http://gdmltest.u-ga.fr/item/BUMI_2010_9_3_1_169_0/

[1] Ford, K. B., The representation of numbers as sums of unlike powers. II. J. Amer. Math. Soc., 9 , no. 4 (1996), 919-940. | MR 1325794 | Zbl 0866.11054

[2] Fuchs, W. H. J. - Wright, E. M., The "easier" Waring problem. Q. J. Math., Oxf. Ser., 10 (1939), 190-209. | MR 408 | Zbl 0022.11501

[3] Hardy, G. H. - Wright, E. M., An introduction to the theory of numbers. Fifth edition. The Clarendon Press, Oxford University Press, New York, 1979. | MR 568909

[4] Jagy, W. C. - Kaplansky, I., Sums of squares, cubes, and higher powers. Experiment. Math., 4 (1995), 169-173. | MR 1387474 | Zbl 0867.11066

[5] Laporta, M. B. S. - Wooley, T. D., The representation of almost all numbers as sums of unlike powers. J. Théor. Nombres Bordeaux 13 (2001), 227-240. | MR 1838083 | Zbl 1048.11074

[6] Roth, K. F., 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

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

[8] Vaughan, R. C., The Hardy-Littlewood method. Cambridge Tracts in Mathematics, 80. Cambridge University Press, Cambridge-New York, 1981. | MR 628618 | Zbl 0455.10034

[9] Wright, E. M., An easier Waring problem. J. London Math. Soc., 9 (1934), 267-272. | MR 1574875 | Zbl 0010.10306