Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.8

A Diophantine System Concerning Sums of Cubes


Zhi Ren
Mission San Jose High School
41717 Palm Avenue
Fremont, CA 94539
USA

Abstract:

We study the Diophantine system

\begin{displaymath}\begin{cases} x_{1}+\cdots+x_{n}=a,\\ x_{1}^3+\cdots+x_{n}^3=b, \end{cases} \end{displaymath}

where $a,b \in \mathbb{Q} ,ab\neq0,n\geq4$, and prove, using the theory of elliptic curves, that it has infinitely many rational parametric solutions depending on n-3 free parameters. Moreover, this Diophantine system has infinitely many positive rational solutions with no common element for n=4, which partially answers a question in our earlier paper.

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Received August 4 2013; revised version received September 4 2013. Published in Journal of Integer Sequences, September 8 2013. Minor revision, November 1 2013.

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