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\(D(-1)\)-triples of the Form \(\{1,b,c\}\) in the Ring \({\mathbb {Z}}[\sqrt{-t}], t>0\)

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Abstract

In this paper, we study \(D(-1)\)-triples of the form \(\{1,b,c\}\) in the ring \({\mathbb {Z}}[\sqrt{-t}]\), \(t>0\), for positive integer b such that b is a prime, twice prime, and twice prime squared. We prove that in those cases, c has to be an integer. In cases of \(b=26,37\), or 50, we prove that \(D(-1)\)-triples of the form \(\{1,b,c\}\) cannot be extended to a \(D(-1)\)-quadruple in the ring \({\mathbb {Z}}[\sqrt{-t}\,], t>0\), except in cases \(t\in \{1,4,9,25,36,49\}\). For those exceptional cases of t we show that there exist infinitely many \(D(-1)\)-quadruples of the form \(\{1,b,-c,d\}\), \(c,d>0\) in \({\mathbb {Z}}[\sqrt{-t}\,]\).

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References

  1. Abu Muriefah, F.S., Al-Rashed, A.: Some Diophantine quadruples in the ring \({\mathbb{Z}}[\sqrt{-2}]\). Math. Commun. 9, 1–8 (2004)

    MathSciNet  MATH  Google Scholar 

  2. Abu Muriefah, F.S., Al-Rashed, A.: On the extendibility of the Diophantine triples \(\{1,5, c\}\). Int. J. Math. Math. Sci. 33, 1737–1746 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baker, A., Wüstholz, G.: Logarithmic forms and group varieties. J. Reine Angew. Math. 498, 19–62 (1993)

    MathSciNet  MATH  Google Scholar 

  4. Bennett, M.A.: On the number of solutions of simultaneous Pell equations. J. Reine Angew. Math 498, 173–199 (1998)

    MathSciNet  MATH  Google Scholar 

  5. Bonciocat, N.C., Cipu, M., Mignotte, M.: On \(D(-1)\)-quadruples. Publ. Mat. 56, 279–304 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dickson, L.E.: History of the Theory of Numbers, vol. 2, pp. 513–520. Chelsea, New York (1996)

    Google Scholar 

  7. Dujella, A.: The problem of Diophantus and Davenport for Gaussian integers. Glas. Mat. Ser III 32, 1–10 (1997)

    MathSciNet  MATH  Google Scholar 

  8. Dujella, A.: Complete solution of a family of simultanous Pellian equations. Acta Math. Inf. Univ. Ostraivensis 6, 59–67 (1998)

    MathSciNet  MATH  Google Scholar 

  9. Dujella, A.: On the exceptional set in the problem of Diophantus and Davenport. Applications of Fibonacci Numbers 7, 69–76 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Dujella, A., Filipin, A., Fuchs, C.: Effective solution of the \(D(-1)\)-quadruple conjecture. Acta Arith. 128, 319–338 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dujella, A., Fuchs, C.: Complete solution of a problem of Diophantus and Euler. J. Lond. Math. Soc. 71, 33–52 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dujella, A., Pethő, A.: Generalization of a theorem of Baker and Davenport. Q. J. Math. Oxf. Ser. 49, 291–306 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dujella, A., Soldo, I.: Diophantine quadruples in \({\mathbb{Z}}[\sqrt{-2}]\). An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 18, 81–98 (2010)

    MathSciNet  MATH  Google Scholar 

  15. Elsholtz, C., Filipin, A., Fujita, Y.: On Diophantine quintuples and \(D(-1)\)-quadruples. Monatsh. Math. (to appear)

  16. Filipin, A.: Nonextendibility of \(D(-1)\)-triples of the form \(\{1,10, c\}\). Int. J. Math. Math. Sci. 14, 2217–2226 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Filipin, A., Fujita, Y.: The number of \(D(-1)\)-quadruples. Math. Commun. 15, 381–391 (2010)

    MathSciNet  MATH  Google Scholar 

  18. Franušić, Z.: On the extensibility of Diophantine triples \(\{k-1, k+1, 4k\}\) for Gaussian integers. Glas. Mat. Ser. III 43, 265–291 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Franušić, Z., Kreso, D.: Nonextensibility of the pair \(\{1,3\}\) to a Diophantine quintuple in \({\mathbb{Z}}[\sqrt{-2}]\). J. Comb. Number Theory 3, 1–15 (2011)

    MathSciNet  MATH  Google Scholar 

  20. Fujita, Y.: The extensibility of \(D(-1)\)-triples \(\{1, b, c\}\). Publ. Math. Debrecen 70, 103–117 (2007)

    MathSciNet  MATH  Google Scholar 

  21. Grelak, A., Grytczuk, A.: On the diophantine equation \(ax^2-by^2 = c\). Publ. Math. Debrecen 44, 191–199 (1994)

    MathSciNet  MATH  Google Scholar 

  22. Ibrahimpašić, B.: A parametric family of quartic Thue inequalities. Bull. Malays. Math. Sci. Soc. 34, 215–230 (2011)

    MathSciNet  MATH  Google Scholar 

  23. Nagell, T.: Introduction to Number Theory. Wiley, New York (1951)

    MATH  Google Scholar 

  24. Smart, N.P.: The Algorithmic Resolution of Diophantine Equations. Cambridge University Press, Cambridge (1998)

    Book  MATH  Google Scholar 

  25. Soldo, I.: On the existence of Diophantine quadruples in \({\mathbb{Z}}[\sqrt{-2}]\). Miskolc Math. Notes 14, 261–273 (2013)

    MathSciNet  MATH  Google Scholar 

  26. Soldo, I.: On the extensibility of \(D(-1)\)-triples \(\{1, b, c\}\) in the ring \({\mathbb{Z}}[\sqrt{-t}], t>0\). Studia Sci. Math. Hung. 50, 296–330 (2013)

    MathSciNet  MATH  Google Scholar 

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Acknowledgments

The author is very grateful to Professor Andrej Dujella for valuable comments and advices. Moreover, the author would like to thank the anonymous referee for carefully reading of the paper and for valuable comments and suggestions that improved the previous version of the paper. This work has been supported by Croatian Science Foundation under the project no. 6422.

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  1. Department of Mathematics, University of Osijek, Trg Ljudevita Gaja 6, 31000, Osijek, Croatia

    Ivan Soldo

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  1. Ivan Soldo
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Correspondence to Ivan Soldo.

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Communicated by V. Ravichandran.

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Soldo, I. \(D(-1)\)-triples of the Form \(\{1,b,c\}\) in the Ring \({\mathbb {Z}}[\sqrt{-t}], t>0\) . Bull. Malays. Math. Sci. Soc. 39, 1201–1224 (2016). https://doi.org/10.1007/s40840-015-0229-7

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  • DOI: https://doi.org/10.1007/s40840-015-0229-7

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