On quartic Diophantine equations with trivial solutions in the Gaussian integers
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,
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- 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.
- ISSN: 1306-6048
- Yayın Aralığı: Yılda 2 Sayı
- Başlangıç: 2007
- Yayıncı: Abdullah HARMANCI