On Waring’s problem: \(g(4)\leq 20\). (English) Zbl 0604.10027

Using the circle method, the author shows that if an integer \(N\) exceeds \(10^{412}\) then it is expressible as a sum of at most 20 fourth powers of integers. He indicates that if \(N>10^{700}\) then 19 fourth powers would suffice.
Computations due to H. E. Thomas jun. [Trans. Am. Math. Soc. 193, 427–430 (1974; Zbl 0294.10033)] indicate that if \(N\leq 10^{412}\) then 20 fourth powers also suffice, thus leading to the result of the title. The accuracy of Thomas’s exponent 412 has been questioned; see e.g. [J. M. Deshouillers, Groupe Étude, Théor. Anal. Nombres, 1984/85, Exp. No. 33 (1985; Zbl 0586.10026)], a point to which the author promises to return.
In the meantime it has been announced that in collaboration with J. M. Deshouillers and F. Dress the author has now shown that 19 fourth powers suffice to represent all \(N\) [see Sémin. Théor. Nombres, Univ. Bordeaux I 1984/85, Exp. No. 14 (1985; Zbl 0586.10027)].
Reviewer: G.Greaves


11P05 Waring’s problem and variants
11P55 Applications of the Hardy-Littlewood method
11D85 Representation problems