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On the Diophantine equation \(a^{3}+b^{3}+c^{3}+d^{3}=0\). (English) Zbl 1031.11015

The equation in the title has been studied by many mathematicians since Diophantus. Partial solutions in integers and complete solutions in rational numbers have been found. The main result shows how “every integral primitive solution \((a,b,c,d)\) can be written uniquely”.

MSC:

11D25 Cubic and quartic Diophantine equations
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[1] Berndt, B. C.; Choi, Y.-S.; Kang, S.-Y., The problems submitted by Ramanujan to the Journal of the Indian Mathematical Society, (Berndt, B. C.; Gesztesy, F., Proceedings of Continued Fractions: From Analytic Number Theory to Constructive Approximation, University of Missouri May 1998. Proceedings of Continued Fractions: From Analytic Number Theory to Constructive Approximation, University of Missouri May 1998, AMS Cont. Math., 236 (1999), Amer. Math. Soc: Amer. Math. Soc Providence), 15-56 · Zbl 1133.11300
[2] Choudhry, A., On equal sums of cubes, Rocky Mountain J. Math., 28, 1251-1257 (1998) · Zbl 0934.11011
[3] Dickson, L. E., History of the Theory of Numbers, Vol. 2, Diophantine Analysis (1966), Chelsea: Chelsea New York · Zbl 0139.26603
[4] Hua, L.-K., Introduction to Number Theory (1982), Springer-Verlag: Springer-Verlag Berlin/New York
[5] Hardy, G. H.; Wright, E. M., An Introduction to the Theory of Numbers (1979), Clarendon Press: Clarendon Press Oxford · Zbl 0423.10001
[6] B. C. Berndt, (Ed.), Ramanujan’s Notebooks, Springer-Verlag, New York, 1985-1998.; B. C. Berndt, (Ed.), Ramanujan’s Notebooks, Springer-Verlag, New York, 1985-1998. · Zbl 0389.10002
[7] Sándor, C., On the equation \(a^3+b^3+c^3=d^3\), Period. Math. Hungar., 33, 121-134 (1996) · Zbl 0880.11029
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