Glasnik Matematicki, Vol. 48, No. 2 (2013), 265-289.

ON A DIOPHANTINE EQUATION OF ANDREJ DUJELLA

Keith R. Matthews, John P. Robertson and Jim White

Department of Mathematics, University of Queensland, Brisbane, Australia, 4072
and
Centre for Mathematics and its Applications, Australian National University, Canberra, ACT, Australia, 0200
e-mail: keithmatt@gmail.com

Actuarial and Economic Services Division, National Council on Compensation Insurance, Boca Raton, FL 33487, USA
e-mail: jpr2718@gmail.com

14 Nash Place, Stirling, Canberra, ACT, Australia, 2611
e-mail: mathimagics@yahoo.co.uk


Abstract.   We investigate positive solutions (x,y) of the Diophantine equation x2-(k2+1)y2=k2 that satisfy y < k-1, where k≥ 2. It has been conjectured that there is at most one such solution for a given k.

2010 Mathematics Subject Classification.   11D09.

Key words and phrases.   Quadratic diophantine equations, continued fractions.


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DOI: 10.3336/gm.48.2.04


References:

  1. L. E. Dickson, Introduction to the theory of numbers, Dover Publications, 1957.

  2. A. Dujella, Continued fractions and RSA with small secret exponent, Tatra Mt. Math. Publ. 29 (2004), 101-112.
    MathSciNet    

  3. A. Dujella and C. Fuchs, Complete solution of a problem of Diophantus and Euler, J. London Math. Soc. 71 (2005), 33-52.
    MathSciNet     CrossRef

  4. A. Dujella and B. Jadrijević, A family of quartic Thue inequalities, Acta Arith. 111 (2004), 61-76.
    MathSciNet     CrossRef

  5. A. Filipin, Y. Fujita and M. Mignotte, The non-extendibility of some parametric families of D(-1)-triples, Q. J. Math. 63 (2012), 605-621.
    MathSciNet     CrossRef

  6. O. Perron, Die Lehre von den Kettenbrüchen, Teubner, Stuttgart, 1954. %
    MathSciNet    

  7. P. Z. Yuan and Z. F. Zhang, On the diophantine equation X2-(1+a2)Y4=-2a, Sci. China Math. 53 (2010), 2143-2158.
    MathSciNet     CrossRef

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