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Waring’s problem for biquadrates: the status in 1984. (Problème de Waring pour les bicarrés: le point en 1984.) (French) Zbl 0586.10026

Groupe Étude Théor. Anal. Nombres 1re/2e Années: 1984/1985, Exp. No. 33, 5 p. (1985).
Waring’s well-known statement concerning fourth powers of integers was that every positive integer is expressible as a sum of at most nineteen of them (the number 19 arising, of course, from the special requirements of rather small integers). While Hardy and Littlewood had shown in 1925 by their (effective) circle method that every sufficiently large number was thus representable, the original statement of Waring remained unverified.
The author announces that Waring’s statement has now been established by R. Balasubramanian, F. Dress and himself and sketches the previous history of the problem. In particular he discusses the thesis of H. E. Thomas where he showed, for example that every integer larger than \(10^{1409}\) or (according to the present author) smaller than \(10^{233}\) is representable in Waring’s form; cf. H. E. Thomas’s paper [Trans. Am. Math. Soc. 193, 427–430 (1974; Zbl 0294.10033)] which is, however, devoted to showing that every positive integer is a sum of at most twenty-two fourth powers of integers.

MSC:

11P05 Waring’s problem and variants
11D72 Diophantine equations in many variables
11D85 Representation problems