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Fourth power diophantine equations in Gaussian integers

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Abstract

In this paper, we examine a class of fourth power diophantine equations of the form \(x^4+kx^2y^2+y^4=z^2\) and \(ax^4+by^4=cz^2\), in the Gaussian integers, where a and b are prime integers.

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References

  1. Cohen H, Number theory, volume I, tools and diophantine equations, Graduate Texts in Math., vol. 239 (2007) (New York: Springer)

  2. Dickson L E, History of the theory of number, volume II, diophantine analysis (1971) (New York: Chelsea Publishing Company)

  3. Mordell L J, Diophantine equations, vol. 30 (1969) (London: Academic Press Inc.)

  4. Najman F, The diophantine equation \(x^4\pm y^4=iz^2\) in the Gaussian integers, Amer. Math. Monthly, 117 (2010) 637–641

    Article  MATH  Google Scholar 

  5. Parker Adam, Who solved the Bernoulli equation and how did they do it?, Coll. Math. J., 44 (2013) 89–97

    Article  MathSciNet  MATH  Google Scholar 

  6. Sage software, Version 4.3.5, http://sagemath.org

  7. Schneiders U and Zimmer H G, The rank of elliptic curves upon quadratic extensions, Computational Number Theory, edited by A. Petho, H. C. Williams and H. G. Zimmer (1991) (Berlin: de Gruyter) pp. 239–260

  8. Suzuki Y, On the diophantine equation \(2^{a}X^{4}+2^{b}Y^{4}=2^{c}Z^{4}\), Proc. Jpn. Acad. Ser. A 72 (1996) 92–94

    Article  Google Scholar 

  9. Suzuki Y, All solutions of diophantine equation \(2^{a}X^{r}+2^{b}Y^{s}=2^{c}Z^{t}\) where \(r, s\) and \(t\) are 2 or 4, Nihonkai Math. J. 7(2) (1996) 113–145

    MathSciNet  MATH  Google Scholar 

  10. Thongjunthug T, Elliptic curves over \(Q(i)\), Honours thesis (2006)

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Acknowledgements

The authors are indebted to the anonymous reviewer of an earlier paper for providing insightful comments and providing directions for additional work which has resulted in this paper.

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Correspondence to Farzali Izadi.

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Communicating Editor: S D Adhikari

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Izadi, F., Rasool, N.F. & Amaneh, A.V. Fourth power diophantine equations in Gaussian integers. Proc Math Sci 128, 18 (2018). https://doi.org/10.1007/s12044-018-0390-7

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  • DOI: https://doi.org/10.1007/s12044-018-0390-7

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2010 Mathematics Subject Classification

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