THE SOLUBILITY OF DIAGONAL CUBIC DIOPHANTINE EQUATIONS

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.