Skip to main content
Log in

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

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

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 \({\mathbb {Z}}[\sqrt{-t}], t>0\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Similar content being viewed by others

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. 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). https://doi.org/10.1007/s00605-017-1061-y

    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. https://oeis.org/A118612. 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 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ivan Soldo.

Additional information

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.01.1.1.01.0004)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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). https://doi.org/10.1007/s40840-018-0638-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-018-0638-5

Keywords

Mathematics Subject Classification

Navigation