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A Pellian Equation with Primes and Applications to \(D(-1)\)-Quadruples

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In this paper, we prove that the equation \(x^2-(p^{2k+2}+1)y^2=-p^{2l+1}\), \(l \in \{0,1,\dots ,k\}, k \ge 0\), where p is an odd prime number, is not solvable in positive integers x and y. By combining that result with other known results on the existence of Diophantine quadruples, we are able to prove results on the extensibility of some \(D(-1)\)-pairs to quadruples in the ring \({\mathbb {Z}}[\sqrt{-t}], t>0\).

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

    Article  MathSciNet  Google Scholar 

  3. Bugeaud, Y., Mignotte, M.: On integers with identical digits. Mathematika 46, 411–417 (1999)

    Article  MathSciNet  Google Scholar 

  4. Crescenzo, P.: A diophantine equation which arises in the theory of finite groups. Adv. Math. 17, 25–29 (1975)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Dujella, A.: Continued fractions and RSA with small secret exponents. Tatra Mt. Math. Publ. 29, 101–112 (2004)

    MathSciNet  MATH  Google Scholar 

  9. Dujella, A.: What is...a Diophantine \(m\)-tuple? Not. Am. Math. Soc. 63, 772–774 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Dujella, A., Jadrijević, B.: A family of quartic Thue inequalities. Acta Arith. 111, 61–76 (2004)

    Article  MathSciNet  Google Scholar 

  12. 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 

  13. Elsholtz, C., Filipin, A., Fujita, Y.: On Diophantine quintuples and \(D(-1)\)-quadruples. Monatsh. Math. 175, 227–239 (2014)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  15. Filipin, A., Fujita, Y., Mignotte, M.: The non-extendibility of some parametric families of \(D(-1)\)- triples. Q. J. Math. 63, 605–621 (2012)

    Article  MathSciNet  Google Scholar 

  16. Fujita, Y.: The non-extensibility of \(D(4k)\)- triples \(\{1, 4k(k-1), 4k^2+1\}\), with \(|k|\) prime. Glas. Mat. Ser. III(41), 205–216 (2006)

    Article  Google Scholar 

  17. 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  Google Scholar 

  18. 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 

  19. Franušić, Z., Soldo, I.: The problem of Diophantus for integers of \({\mathbb{Q}}(\sqrt{-3})\), Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 18, 15–25 (2014)

    MathSciNet  MATH  Google Scholar 

  20. Khosravi, A., Khosravi, B.: A new characterization of some alternating and symmetric groups (II). Houst. J. Math. 30, 953–967 (2004)

    MathSciNet  MATH  Google Scholar 

  21. Lapkova, K.: Explicit upper bound for an average number of divisors of quadratic polynomials. Arch. Math. (Basel) 106, 247–256 (2016)

    Article  MathSciNet  Google Scholar 

  22. Lapkova, K.: Explicit upper bound for the average number of divisors of irreducible quadratic polynomials. Monatsh. Math. (2017).

    Article  MathSciNet  MATH  Google Scholar 

  23. Lapkova, K.: Explicit upper bound for the average number of divisors of irreducible quadratic polynomials. arXiv:1704.02498v3 [math. NT]

  24. Niven, I., Zuckerman, H.S., Montgomery, H.L.: An Introduction to the Theory of Numbers. Wiley, New York (1991)

    MATH  Google Scholar 

  25. Sloane, N.J.A.: The on-line encyclopedia of integer sequences. Accessed 5 Dec 2017

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  28. 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)

    Article  MathSciNet  Google Scholar 

  29. Trudgian, T.: Bounds on the number of Diophantine quintuples. J. Number Theory 157, 233–249 (2015)

    Article  MathSciNet  Google Scholar 

  30. Worley, R.T.: Estimating \(|\alpha -p/q|\). J. Aust. Math. Soc. Ser. A 31, 202–206 (1981)

    Article  Google Scholar 

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Authors are deeply grateful to the anonymous referees for their helpful comments, insights and suggestions that lead to an improved version of the paper.

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Correspondence to Ivan Soldo.

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Communicated by Emrah Kilic.

Authors were supported by the Croatian Science Foundation under the Project No. 6422. A. D. acknowledges support from the QuantiXLie Center of Excellence, a project cofinanced by the Croatian Government and European Union through the European Regional Development Fund—the Competitiveness and Cohesion Operational Programme (Grant KK.

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Dujella, A., Jukić Bokun, M. & Soldo, I. A Pellian Equation with Primes and Applications to \(D(-1)\)-Quadruples. Bull. Malays. Math. Sci. Soc. 42, 2915–2926 (2019).

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