We explain the algorithms that we have implemented to show that all integers congruent to modulo in the interval are sums of five fourth powers, and that all integers congruent to or modulo in the interval are sums of seven fourth powers. We also give some results related to small sums of biquadrates. Combining with the Dickson ascent method, we deduce that all integers in the interval are sums of biquadrates.
Nous expliquons les algorithmes qui nous ont permis de vérifier que tout entier congru à modulo dans l’intervalle est la somme de bicarrés, et que tout entier congru à ou modulo dans l’intervalle est la somme de bicarrés. Nous indiquons également des résultats déduits de calculs portant sur les petites sommes de bicarrés. L’escalade de Dickson appliquée à ces résultats montre que tout entier de l’intervalle est la somme de bicarrés.
@article{JTNB_2000__12_2_411_0, author = {Deshouillers, Jean-Marc and Hennecart, Fran\c{c}ois and Landreau, Bernard}, title = {Waring's problem for sixteen biquadrates. {Numerical} results}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {411--422}, publisher = {Universit\'e Bordeaux I}, volume = {12}, number = {2}, year = {2000}, mrnumber = {1823193}, zbl = {0972.11093}, language = {en}, url = {http://archive.numdam.org/item/JTNB_2000__12_2_411_0/} }
TY - JOUR AU - Deshouillers, Jean-Marc AU - Hennecart, François AU - Landreau, Bernard TI - Waring's problem for sixteen biquadrates. Numerical results JO - Journal de théorie des nombres de Bordeaux PY - 2000 SP - 411 EP - 422 VL - 12 IS - 2 PB - Université Bordeaux I UR - http://archive.numdam.org/item/JTNB_2000__12_2_411_0/ LA - en ID - JTNB_2000__12_2_411_0 ER -
%0 Journal Article %A Deshouillers, Jean-Marc %A Hennecart, François %A Landreau, Bernard %T Waring's problem for sixteen biquadrates. Numerical results %J Journal de théorie des nombres de Bordeaux %D 2000 %P 411-422 %V 12 %N 2 %I Université Bordeaux I %U http://archive.numdam.org/item/JTNB_2000__12_2_411_0/ %G en %F JTNB_2000__12_2_411_0
Deshouillers, Jean-Marc; Hennecart, François; Landreau, Bernard. Waring's problem for sixteen biquadrates. Numerical results. Journal de théorie des nombres de Bordeaux, Volume 12 (2000) no. 2, pp. 411-422. http://archive.numdam.org/item/JTNB_2000__12_2_411_0/
[1] On Waring's problem for fourth powers. Ann. of Math. 40 (1939), 731-747. | JFM | MR | Zbl
,[2] Recent progress on Waring's theorem and its generalizations. Bull. Amer. Math. Soc. 39 (1933), 701-727. | JFM | Zbl
,[3] Problème de Waring pour les bicarrés : le point en 1984. Sém. Théor. Analyt. Nbres Paris, 1984-85, exp. 33. | Numdam | MR | Zbl
,[4] Numerical results for sums of five and seven biquadrates and consequences for sums of 19 biquadrates. Math. Comp. 61, 203 (1993), 195-207. | MR | Zbl
, ,[5] 7 373 170 279 850. Math. Comp. 69 (2000), 421-439. | MR | Zbl
, , ,[6] Bemerkungen zum Waringschen Problem. Math. Ann. 72 (1912), 387-399. | JFM | MR
,[7] A numerical approach to Waring's problem for fourth powers. Ph.D., The University of Michigan, 1973.
,[8] Waring's problem for twenty-two biquadrates. Trans. Amer. Math. Soc. 193 (1974), 427-430. | MR | Zbl
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