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Pellian equations of special type

  • Mirela Jukić Bokun EMAIL logo and Ivan Soldo
Published/Copyright: December 10, 2021
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 71 Issue 6

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
Keywords: Pellian equation; Diophantine m-tuple; quadratic field

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), 1–8.10.1155/S0161171204305181Search 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), 78–86.10.1090/S0025-5718-1959-0105784-2Search 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), 296–330.10.1556/sscmath.50.2013.3.1244Search 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.: Estimating ∣α − p/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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  3. Composition of binary quadratic forms over number fields
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https://doi.org/10.1515/ms-2021-0074
Keywords for this article
Pellian equation; Diophantine m-tuple; quadratic field

Articles in the same Issue

  1. Regular Papers
  2. Semidistributivity and Whitman Property in implication zroupoids
  3. Composition of binary quadratic forms over number fields
  4. On Z-Symmetric Rings
  5. On the factorable spaces of absolutely p-summable, null, convergent, and bounded sequences
  6. Coefficient estimates for Libera type bi-close-to-convex functions
  7. Oscillation of nonlinear third-order differential equations with several sublinear neutral terms
  8. On rapidly oscillating solutions of a nonlinear elliptic equation
  9. Multiplicity of solutions for a class of fourth-order elliptic equations of p(x)-Kirchhoff type
  10. Existence of traveling wave solutions in nonlocal delayed higher-dimensional lattice systems with quasi-monotone nonlinearities
  11. On absolute double summability methods with high indices
  12. On the continuity of lattice isomorphisms on C(X, I)
  13. New fixed point theorems for countably condensing maps with an application to functional integral inclusions
  14. Common fixed point results under w-distance with applications to nonlinear integral equations and nonlinear fractional Differential Equations
  15. The form of locally defined operators in waterman spaces
  16. Conformal vector fields on almost co-Kähler manifolds
  17. A certain η-parallelism on real hypersurfaces in a nonflat complex space form
  18. On log-bimodal alpha-power distributions with application to nickel contents and erosion data
  19. Univariate and bivariate extensions of the generalized exponential distributions
  20. Pellian equations of special type
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