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THE SOLUBILITY OF DIAGONAL CUBIC DIOPHANTINE EQUATIONS
Published online by Cambridge University Press: 01 September 1999
Abstract
This paper proves conditional existence results for non-trivial solutions of the equation
\begin{equation} \sum_{i=1}^{n}a_{i}X_{i}^{3}=0 \quad (n=4\mbox{ or }5), \tag{$*$} \end{equation}
where the coefficients $a_{i}$ and the unknowns $X_{i}$ are taken to be rational integers.
No such results were previously known for $n\leq 6$. The proofs use elementary facts about the 3-descent procedure for elliptic curves of the form $E_{A}: X^{3}+Y^{3}=AZ^{3}$.
Thus, when $n=4$, and the $a_{i}$ are each prime, and are all congruent to 2 modulo 3, it is shown that ($*$) will have non-trivial solutions, providing that the Selmer conjecture holds for the curves $E_{A}$. One may replace the Selmer conjecture by an appropriate form of the Generalized Riemann Hypothesis, when $n=5$ and the $a_{i}$ are again taken to be primes, all congruent to 8 modulo 9. Finally, when $n=5$, one may require only that the $a_{i}$ be square-free and coprime to 3, providing one assumes both the Selmer conjecture and a special case of Schinzel's conjecture (on the representation of primes by cubic polynomials).
1991 Mathematics Subject Classification: 11D25, 11G05, 14G05.
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- Research Article
- Information
- Proceedings of the London Mathematical Society , Volume 79 , Issue 2 , September 1999 , pp. 241 - 259
- Copyright
- 1999 London Mathematical Society
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