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On quartic Diophantine equations with trivial solutions in the Gaussian integers

Year 2022, Volume: 31 Issue: 31, 134 - 142, 17.01.2022
https://doi.org/10.24330/ieja.964819

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.

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.
Year 2022, Volume: 31 Issue: 31, 134 - 142, 17.01.2022
https://doi.org/10.24330/ieja.964819

Abstract

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.
There are 9 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Mahnaz Ahmadı This is me

Ali S. Janfada This is me

Publication Date January 17, 2022
Published in Issue Year 2022 Volume: 31 Issue: 31

Cite

APA Ahmadı, M., & Janfada, A. S. (2022). On quartic Diophantine equations with trivial solutions in the Gaussian integers. International Electronic Journal of Algebra, 31(31), 134-142. https://doi.org/10.24330/ieja.964819
AMA Ahmadı M, Janfada AS. On quartic Diophantine equations with trivial solutions in the Gaussian integers. IEJA. January 2022;31(31):134-142. doi:10.24330/ieja.964819
Chicago Ahmadı, Mahnaz, and Ali S. Janfada. “On Quartic Diophantine Equations With Trivial Solutions in the Gaussian Integers”. International Electronic Journal of Algebra 31, no. 31 (January 2022): 134-42. https://doi.org/10.24330/ieja.964819.
EndNote Ahmadı M, Janfada AS (January 1, 2022) On quartic Diophantine equations with trivial solutions in the Gaussian integers. International Electronic Journal of Algebra 31 31 134–142.
IEEE M. Ahmadı and A. S. Janfada, “On quartic Diophantine equations with trivial solutions in the Gaussian integers”, IEJA, vol. 31, no. 31, pp. 134–142, 2022, doi: 10.24330/ieja.964819.
ISNAD Ahmadı, Mahnaz - Janfada, Ali S. “On Quartic Diophantine Equations With Trivial Solutions in the Gaussian Integers”. International Electronic Journal of Algebra 31/31 (January 2022), 134-142. https://doi.org/10.24330/ieja.964819.
JAMA Ahmadı M, Janfada AS. On quartic Diophantine equations with trivial solutions in the Gaussian integers. IEJA. 2022;31:134–142.
MLA Ahmadı, Mahnaz and Ali S. Janfada. “On Quartic Diophantine Equations With Trivial Solutions in the Gaussian Integers”. International Electronic Journal of Algebra, vol. 31, no. 31, 2022, pp. 134-42, doi:10.24330/ieja.964819.
Vancouver Ahmadı M, Janfada AS. On quartic Diophantine equations with trivial solutions in the Gaussian integers. IEJA. 2022;31(31):134-42.