Solution of certain Diophantine equations in Gaussian integers

TL;DR

The article analyzes quartic Diophantine equations $x^4 \pm pq y^4 = \pm z^2$ and $x^4 \pm pq y^4 = \pm iz^2$ over the Gaussian integers for primes satisfying $p \equiv 3 \pmod 8$, $q \equiv 1 \pmod 8$, and $(\frac{p}{q}) = -1$. By formulating the problem in terms of elliptic curves over $\mathbb{Q}(i)$, the authors compute the ranks and torsion structures of two curve families $E_{pq}^{+}$ and $E_{pq}^{-}$ using $2$-descent and the extended Nagell–Lutz criterion, proving that both have rank $0$ and torsion $\mathbb{Z}/2\mathbb{Z}$ over $\mathbb{Q}(i)$. They then show that any nontrivial Gaussian-integer solution would yield a nontrivial rational point on these rank-zero curves, a contradiction, thereby establishing that the given quartic equations have only trivial solutions in $\mathbb{Z}[i]$. The authors also extend the result to equations with factors $2^n$ via parity arguments, producing a corollary that broadens the nonexistence of nontrivial solutions in the Gaussian setting. Overall, the work demonstrates how elliptic-curve methods and descent techniques in $\mathbb{Q}(i)$ can decisively constrain Diophantine equations of quartic form.

Abstract

In this article, we show that the quartic Diophantine equations $x^4 \pm pqy^4=\pm z^2$ and $ x^4 \pm pq y^4= \pm iz^2$ have only trivial solutions for some primes $p$ and $q$ satisfying conditions $ p \equiv 3 \pmod 8, ~ q \equiv 1 \pmod 8 ~\text{and}~ \displaystyle\legendre{p}{q} = -1$. Here we have found the torsion of the two families of elliptic curves to find the solutions of given Diophantine equations. Moreover, we also calculate the rank of these two families of elliptic curves over the Gaussian field $\mathbb{Q}(i)$.

Theorems & Definitions (14)

• Theorem 1.1
• Corollary 1.1
• Remark 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1
• Proposition 2.1
• Theorem 2.2
• Theorem 2.3
• proof