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Diophantine quadruples in \(\mathbb {Z}[i][X]\)

Periodica Mathematica Hungarica volume 82, pages 198–212 (2021)Cite this article

Abstract

In this paper, we prove that every Diophantine quadruple in \(\mathbb {Z}[i][X]\) is regular. More precisely, we prove that if \(\{a, b, c, d\}\) is a set of four non-zero polynomials from \(\mathbb {Z}[i][X]\), not all constant, such that the product of any two of its distinct elements increased by 1 is a square of a polynomial from \(\mathbb {Z}[i][X]\), then

$$\begin{aligned} (a+b-c-d)^2=4(ab+1)(cd+1). \end{aligned}$$

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Notes

  1. In an (improper) D(1)-quadruple, we cannot have two equal non-constant polynomials because then it would be for example \((b-w)(b+w)=-1\) which is not possible since both factors on the left hand side of this equation cannot be constant.

References

  1. N. Adžaga, On the size of Diophantine m-tuples in imaginary quadratic number rings. Bull. Math. Sci. 9(3), 1950020 (2019). (10 pages)

    MathSciNet  Google Scholar 

  2. J. Arkin, V.E. Hoggatt, E.G. Strauss, On Euler’s solution of a problem of Diophantus. Fibonacci Quart. 17, 333–339 (1979)

    MathSciNet  MATH  Google Scholar 

  3. M. Bliznac Trebješanin, A. Filipin, A. Jurasić, On the polynomial quadruples with the property \(D(-1;1)\). Tokyo J. Math. 41(2), 527–540 (2018)

    Article  MathSciNet  Google Scholar 

  4. E. Brown, Sets in which \(xy + k\) is always a square. Math. Comput. 45, 613–620 (1985)

    MathSciNet  MATH  Google Scholar 

  5. M. Cipu, Y. Fujita, T. Miyazaki, On the number of extensions of a Diophantine triple. Int. J. Number Theory 14, 899–917 (2018)

    Article  MathSciNet  Google Scholar 

  6. A. Dujella, Diophantine\(m\)-tuples. http://web.math.pmf.unizg.hr/~duje/dtuples.html. Accessed 15 Jan 2019

  7. A. Dujella, Generalization of a problem of Diophantus. Acta Arith. 65, 15–27 (1993)

    Article  MathSciNet  Google Scholar 

  8. A. Dujella, On the size of Diophantine \(m\)-tuples. Math. Proc. Camb. Philos. Soc. 132, 23–33 (2002)

    Article  MathSciNet  Google Scholar 

  9. A. Dujella, Bounds for the size of sets with the property \(D(n)\). Glas. Mat. Ser. III(39), 199–205 (2004)

    Article  MathSciNet  Google Scholar 

  10. A. Dujella, There are only finitely many Diophantine quintuples. J. Reine Angew. Math. 566, 183–214 (2004)

    MathSciNet  MATH  Google Scholar 

  11. A. Dujella, C. Fuchs, A polynomial variant of a problem of Diophantus and Euler. Rocky Mt. J. Math. 33, 797–811 (2003)

    Article  MathSciNet  Google Scholar 

  12. A. Dujella, C. Fuchs, Complete solution of the polynomial version of a problem of Diophantus. J. Number Theory 106, 326–344 (2004)

    Article  MathSciNet  Google Scholar 

  13. A. Dujella, C. Fuchs, F. Luca, A polynomial variant of a problem of Diophantus for pure powers. Int. J. Number Theory 4, 57–71 (2008)

    Article  MathSciNet  Google Scholar 

  14. A. Dujella, C. Fuchs, R.F. Tichy, Diophantine \(m\)-tuples for linear polynomials. Period. Math. Hung. 45, 21–33 (2002)

    Article  MathSciNet  Google Scholar 

  15. A. Dujella, C. Fuchs, G. Walsh, Diophantine \(m\)-tuples for linear polynomials. II. Equal degrees. J. Number Theory 120, 213–228 (2006)

    Article  MathSciNet  Google Scholar 

  16. A. Dujella, A. Jurasić, On the size of sets in a polynomial variant of a problem of Diophantus. Int. J. Number Theory 6, 1449–1471 (2010)

    Article  MathSciNet  Google Scholar 

  17. A. Dujella, F. Luca, On a problem of Diophantus with polynomials. Rocky Mt. J. Math. 37, 131–157 (2007)

    MathSciNet  MATH  Google Scholar 

  18. A. Filipin, A. Jurasić, On the size of Diophantine \(m\)-tuples for linear polynomials. Miskolc Math. Notes 17(2), 861–876 (2016)

    Article  MathSciNet  Google Scholar 

  19. A. Filipin, A. Jurasić, A polynomial variant of a problem of Diophantus and its consequences. Glas. Mat. Ser. III 54, 21–52 (2019)

  20. P. Gibbs, A generalised Stern–Brocot tree from regular Diophantine quadruples. XXX Mathematics Archive. arXiv:math/9903035 [math.NT]

  21. P. Gibbs, Some rational sextuples. Glas. Mat. Ser. III(41), 195–203 (2006)

    Article  MathSciNet  Google Scholar 

  22. B. He, A. Togbé, V. Ziegler, There is no Diophantine quintuple. Trans. Am. Math. Soc. 371, 6665–6709 (2019)

    Article  MathSciNet  Google Scholar 

  23. B.W. Jones, A variation of a problem of Davenport and Diophantus. Quart. J. Math. Oxf. Ser. (2) 27, 349–353 (1976)

    Article  MathSciNet  Google Scholar 

  24. B.W. Jones, A second variation of a problem of Davenport and Diophantus. Fibonacci Quart. 15, 323–330 (1977)

    MathSciNet  Google Scholar 

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Acknowledgements

The authors were supported by the Croatian Science Foundation under the Project No. IP-2018-01-1313.

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Authors and Affiliations

  1. Faculty of Civil Engineering, University of Zagreb, Fra Andrije Kačića-Miošića 26, 10000, Zagreb, Croatia

    Alan Filipin

  2. Department of Mathematics, University of Rijeka, Radmile Matejčić 2, 51000, Rijeka, Croatia

    Ana Jurasić

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  1. Alan Filipin
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  2. Ana Jurasić
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Correspondence to Ana Jurasić.

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Filipin, A., Jurasić, A. Diophantine quadruples in \(\mathbb {Z}[i][X]\). Period Math Hung 82, 198–212 (2021). https://doi.org/10.1007/s10998-020-00353-y

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  • DOI: https://doi.org/10.1007/s10998-020-00353-y

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