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Pellian equations of special type

  • Mirela Jukić Bokun EMAIL logo and Ivan Soldo
From the journal Mathematica Slovaca

Abstract

In this paper, we consider the solvability of the Pellian equation

x2(d2+1)y2=m,

in cases d = nk, m = n2l−1, where k, l are positive integers, n is a composite positive integer and d = pq, m = pq2, p, q are primes. We use the obtained results to prove results on the extendibility of some D(−1)-pairs to quadruples in the ring Z[t] , with t > 0.

MSC 2010: 11D09; 11R11

The second author was supported by the Croatian Science Foundation under the project no. IP-2018-01-1313.


Acknowledgement

The authors would like to thank the referees for their helpful remarks and suggestions which improved the first version of the paper.

  1. ((Communicated by István Gaál )

References

[1] Abu Muriefah, F. S.—Al-Rashed, A.: Some Diophantine quadruples in the ring Z[2] , Math. Commun. 9 (2004), 1–8.Search in Google Scholar

[2] Bayad, A.—Dossavi-Yovo, A.—Filipin, A.—Togbe, A.: On the extensibility of D(4)-triple {k−2, k+2, 4k} over Gaussian integers, Notes Number Theory Discrete Math. 23 (2017), 1–26.Search in Google Scholar

[3] Deza, E.—Deza, M. M.: Figurate Numbers, World Scientific, Singapore, 2012.10.1142/8188Search in Google Scholar

[4] Dujella, A., The problem of Diophantus and Davenport for Gaussian integers, Glas. Mat. Ser III 32 (1997), 1–10.Search in Google Scholar

[5] Dujella, A., Continued fractions and RSA with small secret exponents, Tatra Mt. Math. Publ. 29 (2004), 101–112.Search in Google Scholar

[6] Dujella, A.—Jadrijević, B.: A family of quartic Thue inequalities, Acta Arith. 111 (2004), 61–76.10.4064/aa111-1-5Search in Google Scholar

[7] Dujella, A.—Soldo, I.: Diophantine quadruples in Z[2] , An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 18 (2010), 81–98.Search in Google Scholar

[8] Dujella, A.—Jukić Bokun, M.—Soldo, I.: A Pellian equation with primes and applications to D(−1)-quadruples Bull. Malays. Math. Sci. Soc. 42 (2019), 2915–2926.10.1007/s40840-018-0638-5Search in Google Scholar

[9] Filipin, A.—Fujita, Y.—Mignotte, M.: The non-extendibility of some parametric families of D(−1)-triples, Quart. J. Math. 63 (2012), 605–621.10.1093/qmath/har013Search in Google Scholar

[10] Filipin, A.—Jukić Bokun, M.—Soldo, I.: On D(−1)-triples {1, 4p2 +1, 1−p} in the ring Z[p] with a prime p, Period. Math. Hungar., to appear.Search in Google Scholar

[11] Franušić, Z.—Kreso, D.: Nonextensibility of the pair {1, 3} to a Diophantine quintuple in Z[2] , J. Comb. Number Theory 3 (2011), 1–15.Search in Google Scholar

[12] Jukić Bokun, M.—Soldo, I.: On the extensibility of D(−1)-pairs containing Fermat primes, Acta Math. Hungar. 159 (2019), 89–108.10.1007/s10474-019-00951-4Search in Google Scholar

[13] Nagell, T.: Introduction to Number Theory, Wiley, New York, 1951.Search in Google Scholar

[14] Niven, I.—Zuckerman, H. S.—Montgomery, H. L.: An Introduction to the Theory of Numbers, John Wiley & Sons, New York, 1991.Search in Google Scholar

[15] Shanks, D.: A sieve method for factoring numbers of the form n2 + 1, MTAC 13 (1959), 78–86.Search in Google Scholar

[16] Shanks, D.: An analytic criterion for the existence of infinitely many primes of the form 12 , Illinois J. Math. 8 (1964), 377–379.10.1215/ijm/1256059560Search in Google Scholar

[17] Sloane, N. J. A.: The on-line encyclopedia of integer sequences, available at https://oeis.org/.Search in Google Scholar

[18] Soldo, I.: On the existence of Diophantine quadruples in Z[2] , Miskolc Math. Notes 14 (2013), 261–273.10.18514/MMN.2013.565Search in Google Scholar

[19] Soldo, I.: On the extensibility of D(−1)-triples {1, b, c} in the ring Z[t] , t > 0, Studia Sci. Math. Hungar. 50 (2013), 296–330.Search in Google Scholar

[20] Soldo, I.: D(−1)-triples of the form {1, b, c} in the ring Z[t] , t > 0, Bull. Malays. Math. Sci. Soc. 39 (2016), 1201–1224.10.1007/s40840-015-0229-7Search in Google Scholar

[21] Worley, R. T.: Estimatingαp/q∣, J. Austral. Math. Soc. Ser. A 31 (1981), 202–206.10.1017/S1446788700033486Search in Google Scholar

Received: 2020-06-17
Accepted: 2020-12-21
Published Online: 2021-12-10
Published in Print: 2021-12-20

© 2021 Mathematical Institute Slovak Academy of Sciences

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