Deshouillers, Jean-Marc; Hennecart, François; Landreau, Bernard Waring’s problem for sixteen biquadrates – numerical results. (English) Zbl 0972.11093 J. Théor. Nombres Bordx. 12, No. 2, 411-422 (2000). H. Davenport [Ann. Math (2) 40, 731-747 (1939; Zbl 0024.01402)] showed that all large numbers are sums of 16 biquadrates. More recently, H. E. Thomas [Trans. Am. Math. Soc. 193, 427-430 (1974; Zbl 0294.10033)] showed that every number in the interval \([13793,10^{80}]\) is a sum of 16 biquadrates. Here the authors extend the upper end of the interval to \(10^{245}\). It is also announced that in a forthcoming paper, the first author together with K. Kawada and T. D. Wooley, has established that every number exceeding \(10^{220}\) is a sum of 16 biquadrates. The list of the 96 numbers up to 13792 which require at least 17 biquadrates is also given. Reviewer: Peter Shiu (Loughborough) Cited in 1 ReviewCited in 2 Documents MSC: 11P05 Waring’s problem and variants 11Y99 Computational number theory Keywords:Waring problem; sums of 16 biquadrates Citations:Zbl 0024.01402; Zbl 0294.10033 × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML EMIS Online Encyclopedia of Integer Sequences: Sum of 19 but no fewer nonzero fourth powers. Numbers that are the sum of no fewer than 17 biquadrates (4th powers). Smallest k such that n^4 = a_1^4+...+a_k^4 and all a_i are positive integers less than n. Total number of positive integers below 10^n requiring 19 positive biquadrates in their representation as sum of biquadrates. References: [1] Davenport, H., On Waring’s problem for fourth powers. Ann. of Math.40 (1939), 731-747. · JFM 65.1149.02 [2] Dickson, L.E., Recent progress on Waring’s theorem and its generalizations. Bull. Amer. Math. Soc.39 (1933), 701-727. · JFM 59.0177.01 [3] Deshouillers, J-M., Problème de Waring pour les bicarrés : le point en 1984. Sém. Théor. Analyt. Nbres Paris, 1984-85, exp. 33. · Zbl 0586.10026 [4] Deshouillers, J-M., Dress, F., Numerical results for sums of five and seven biquadrates and consequences for sums of 19 biquadrates. Math. Comp.61, 203 (1993), 195-207. · Zbl 0879.11052 [5] Deshouillers, J-M., Hennecart, F., Landreau, B., 7 373 170 279 850. Math. Comp.69 (2000), 421-439. · Zbl 0937.11061 [6] Kempner, A., Bemerkungen zum Waringschen Problem. Math. Ann.72 (1912), 387-399. · JFM 43.0239.02 [7] Thomas, H.E., A numerical approach to Waring’s problem for fourth powers. Ph.D., The University of Michigan, 1973. [8] Thomas, H.E., Waring’s problem for twenty-two biquadrates. Trans. Amer. Math. Soc.193 (1974), 427-430. · Zbl 0294.10033 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.