On quartic Diophantine equations with trivial solutions in the Gaussian integers

Year 2022, Volume: 31 Issue: 31, 134 - 142, 17.01.2022

Mahnaz Ahmadı Ali S. Janfada

https://doi.org/10.24330/ieja.964819

Cited By: 1

Abstract

We show that the quartic Diophantine equations $ax^4+by^4=cz^2$ has only trivial solution in the Gaussian integers
for some particular choices of $a,b$ and $c$. Our strategy is by elliptic curves method. In fact, we exhibit
two null-rank corresponding families of elliptic curves over Gaussian field. We also determine the torsion groups of both families.

Keywords

Diophantine equation, elliptic curves, quartic equation, number of solutions of Diophantine equations

References

• A. Bremner and J. W. S. Cassels, On the equation $y^2 = X(X^2+p)$, Math. Comp., 42 (165) (1984), 257-264.
• H. Cohen, Number theory. Volume I: Tools and Diophantine equations, Springer, 2007.
• D. Hilbert, Die Theorie der algebraischen Zahlkorper, Jahresbericht der Deutschen Mathematiker-Vereinigung, 4 (1894/95), 175-535.
• F. Izadi, R. F. Naghdali and P. G. Brown, Some quadratic Diophantine equation in the Gaussian integers, Bull. Aust. Math. Soc., 92 (2) (2015), 187-194.
• F. Najman, The Diophantine equation$x^4 \pm y^4 = iz^2$ in Gaussian integers, Amer. Math. Monthly, 117 (7) (2010), 637-641.
• F. Najman, Torsion of elliptic curves over quadratic cyclotomic fields, Math.J.Okayama Univ., 53 (2011), 75-82.
• S. Szabo, Some fourth degree Diophantine equations in Gaussian integers, Integers, 4:paper a16, 17, 2004.
• J. H. Silverman, The arithmetic of elliptic curves, 2nd edition, Springer, 2009.
• J. H. Silverman and J. T. Tate, Rational points on elliptic curves, 2nd edition, Springer, 2015.