# Waring's problem for sixteen biquadrates. Numerical results

• Volume: 12, Issue: 2, page 411-422
• ISSN: 1246-7405

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## Abstract

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We explain the algorithms that we have implemented to show that all integers congruent to $4$ modulo $80$ in the interval $\left[6×{10}^{12}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{0.166667em}{0ex}}2.17×{10}^{14}\right]$ are sums of five fourth powers, and that all integers congruent to $6,21$ or $36$ modulo $80$ in the interval $\left[6×{10}^{12}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{0.166667em}{0ex}}1.36×{10}^{23}\right]$ 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 $\left[13793\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{0.166667em}{0ex}}{10}^{245}\right]$ are sums of $16$ biquadrates.

## How to cite

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Deshouillers, Jean-Marc, Hennecart, François, and Landreau, Bernard. "Waring's problem for sixteen biquadrates. Numerical results." Journal de théorie des nombres de Bordeaux 12.2 (2000): 411-422. <http://eudml.org/doc/248502>.

@article{Deshouillers2000,
abstract = {We explain the algorithms that we have implemented to show that all integers congruent to $4$ modulo $80$ in the interval $[6 \times 10^\{12\} \,;\, 2.17 \times 10^\{14\}]$ are sums of five fourth powers, and that all integers congruent to $6, 21$ or $36$ modulo $80$ in the interval $[6 \times 10^\{12\} \, ;\, 1.36 \times 10^\{23\}]$ 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.},
author = {Deshouillers, Jean-Marc, Hennecart, François, Landreau, Bernard},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Waring problem; sums of 16 biquadrates},
language = {eng},
number = {2},
pages = {411-422},
publisher = {Université Bordeaux I},
title = {Waring's problem for sixteen biquadrates. Numerical results},
url = {http://eudml.org/doc/248502},
volume = {12},
year = {2000},
}

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
PB - Université Bordeaux I
VL - 12
IS - 2
SP - 411
EP - 422
AB - We explain the algorithms that we have implemented to show that all integers congruent to $4$ modulo $80$ in the interval $[6 \times 10^{12} \,;\, 2.17 \times 10^{14}]$ are sums of five fourth powers, and that all integers congruent to $6, 21$ or $36$ modulo $80$ in the interval $[6 \times 10^{12} \, ;\, 1.36 \times 10^{23}]$ 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.
LA - eng
KW - Waring problem; sums of 16 biquadrates
UR - http://eudml.org/doc/248502
ER -

## References

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1. [1] H. Davenport, On Waring's problem for fourth powers. Ann. of Math.40 (1939), 731-747. Zbl0024.01402MR253JFM65.1149.02
2. [2] L.E. Dickson, Recent progress on Waring's theorem and its generalizations. Bull. Amer. Math. Soc.39 (1933), 701-727. Zbl0008.00501JFM59.0177.01
3. [3] J-M. Deshouillers, Problème de Waring pour les bicarrés : le point en 1984. Sém. Théor. Analyt. Nbres Paris, 1984-85, exp. 33. Zbl0586.10026MR849015
4. [4] J-M. Deshouillers, F. Dress, Numerical results for sums of five and seven biquadrates and consequences for sums of 19 biquadrates. Math. Comp.61, 203 (1993), 195-207. Zbl0879.11052MR1201766
5. [5] J-M. Deshouillers, F. Hennecart, B. Landreau, 7 373 170 279 850. Math. Comp.69 (2000), 421-439. Zbl0937.11061MR1651751
6. [6] A. Kempner, Bemerkungen zum Waringschen Problem. Math. Ann.72 (1912), 387-399. Zbl43.0239.02MR1511703JFM43.0239.02
7. [7] H.E. Thomas, A numerical approach to Waring's problem for fourth powers. Ph.D., The University of Michigan, 1973.
8. [8] H.E. Thomas, Waring's problem for twenty-two biquadrates. Trans. Amer. Math. Soc.193 (1974), 427-430. Zbl0294.10033MR342478

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