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 \({\mathbb {Z}}[\sqrt{-t}], t>0\).
Similar content being viewed by others
References
Abu Muriefah, F.S., Al-Rashed, A.: Some diophantine quadruples in the ring \({\mathbb{Z}}[\sqrt{-2}]\). Math. Commun. 9, 1–8 (2004)
Bonciocat, N.C., Cipu, M., Mignotte, M.: On \(D(-1)\)-quadruples. Publ. Mat. 56, 279–304 (2012)
Bugeaud, Y., Mignotte, M.: On integers with identical digits. Mathematika 46, 411–417 (1999)
Crescenzo, P.: A diophantine equation which arises in the theory of finite groups. Adv. Math. 17, 25–29 (1975)
Dujella, A.: The problem of Diophantus and Davenport for Gaussian integers. Glas. Mat. Ser. III(32), 1–10 (1997)
Dujella, A.: On the exceptional set in the problem of Diophantus and Davenport. Appl. Fibonacci Numbers 7, 69–76 (1998)
Dujella, A.: On the size of Diophantine m-tuples. Math. Proc. Camb. Philos. Soc. 132, 23–33 (2002)
Dujella, A.: Continued fractions and RSA with small secret exponents. Tatra Mt. Math. Publ. 29, 101–112 (2004)
Dujella, A.: What is...a Diophantine \(m\)-tuple? Not. Am. Math. Soc. 63, 772–774 (2016)
Dujella, A., Filipin, A., Fuchs, C.: Effective solution of the \(D(-1)\)-quadruple conjecture. Acta Arith. 128, 319–338 (2007)
Dujella, A., Jadrijević, B.: A family of quartic Thue inequalities. Acta Arith. 111, 61–76 (2004)
Dujella, A., Soldo, I.: Diophantine quadruples in \({\mathbb{Z}}[\sqrt{-2}\,]\). An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 18, 81–98 (2010)
Elsholtz, C., Filipin, A., Fujita, Y.: On Diophantine quintuples and \(D(-1)\)-quadruples. Monatsh. Math. 175, 227–239 (2014)
Filipin, A., Fujita, Y.: The number of \(D(-1)\)-quadruples. Math. Commun. 15, 381–391 (2010)
Filipin, A., Fujita, Y., Mignotte, M.: The non-extendibility of some parametric families of \(D(-1)\)- triples. Q. J. Math. 63, 605–621 (2012)
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)
Franušić, Z.: On the extensibility of Diophantine triples \(\{k-1, k+1, 4k\}\) for Gaussian integers. Glas. Mat. Ser. III(43), 265–291 (2008)
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)
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)
Khosravi, A., Khosravi, B.: A new characterization of some alternating and symmetric groups (II). Houst. J. Math. 30, 953–967 (2004)
Lapkova, K.: Explicit upper bound for an average number of divisors of quadratic polynomials. Arch. Math. (Basel) 106, 247–256 (2016)
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
Lapkova, K.: Explicit upper bound for the average number of divisors of irreducible quadratic polynomials. arXiv:1704.02498v3 [math. NT]
Niven, I., Zuckerman, H.S., Montgomery, H.L.: An Introduction to the Theory of Numbers. Wiley, New York (1991)
Sloane, N.J.A.: The on-line encyclopedia of integer sequences. https://oeis.org/A118612. Accessed 5 Dec 2017
Soldo, I.: On the existence of Diophantine quadruples in \({\mathbb{Z}}[\sqrt{-2}\,]\). Miskolc Math. Notes 14, 261–273 (2013)
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)
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)
Trudgian, T.: Bounds on the number of Diophantine quintuples. J. Number Theory 157, 233–249 (2015)
Worley, R.T.: Estimating \(|\alpha -p/q|\). J. Aust. Math. Soc. Ser. A 31, 202–206 (1981)
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
Corresponding author
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
About this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-018-0638-5