Waring's problem for sixteen biquadrates. Numerical results
Journal de théorie des nombres de Bordeaux, Volume 12 (2000) no. 2, p. 411-422

We explain the algorithms that we have implemented to show that all integers congruent to 4 modulo 80 in the interval [6×10 12 ;2.17×10 14 ] are sums of five fourth powers, and that all integers congruent to 6,21 or 36 modulo 80 in the interval [6×10 12 ;1.36×10 23 ] 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 [13793;10 245 ] are sums of 16 biquadrates.

Nous expliquons les algorithmes qui nous ont permis de vérifier que tout entier congru à 4 modulo 80 dans l’intervalle [6×10 12 ;2.17×10 14 ] est la somme de 5 bicarrés, et que tout entier congru à 6,21 ou 36 modulo 80 dans l’intervalle [6×10 12 ;1.36×10 23 ] est la somme de 7 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 [13793;10 245 ] est la somme de 16 bicarrés.

@article{JTNB_2000__12_2_411_0,
     author = {Deshouillers, Jean-Marc and Hennecart, Fran\c cois and Landreau, Bernard},
     title = {Waring's problem for sixteen biquadrates. Numerical results},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {12},
     number = {2},
     year = {2000},
     pages = {411-422},
     zbl = {0972.11093},
     mrnumber = {1823193},
     language = {en},
     url = {http://www.numdam.org/item/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://www.numdam.org/item/JTNB_2000__12_2_411_0/

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