Glasnik Matematicki, Vol. 47, No. 1 (2012), 53-59.

A NOTE ON THE SIMULTANEOUS PELL EQUATIONS X2-AY2=1 AND Z2-BY2=1

Maohua Le

Department of Mathematics, Zhanjiang Normal College, Zhanjiang, Guangdong 524048, P.R. China
e-mail: lemaohua2008@163.com


Abstract.   Let m,n be positive integers with 1 < m < n. Let δ be a positive number with 1/2 < δ < 1 . In this paper we prove that if gcd(m,n)>nδ and n>(8× 1016(log(10163))33)1/θ, where θ=min(1-δ, 2δ-1), then the simultaneous Pell equations x2-(m2-1)y2=1 and z2-(n2-1)y2=1 have only one positive integer solution (x,y,z)=(m,1,n).

2010 Mathematics Subject Classification.   11D09.

Key words and phrases.   Simultaneous Pell equations; number of solutions.


Full text (PDF) (free access)

DOI: 10.3336/gm.47.1.04


References:

  1. M. A. Bennett, Solving families of simultaneous Pell equations, J. Number Theory 67 (1997), 246-251.
    MathSciNet     CrossRef

  2. M. A. Bennett, On the number of solutions of simultaneous Pell equations, J. Reine Angew. Math. 498 (1998), 173-199.
    MathSciNet     CrossRef

  3. M. A. Bennett, M. Cipu, M. Mignotte and R. Okazaki, On the number of solutions of simultaneous Pell equations, Acta Arith. 122 (2006), 407-417.
    MathSciNet     CrossRef

  4. M.-H. Le, On the simultaneous Pell equations x2-D1y2 and z2-D2y2, Adv. Math. China 30 (2001), 87-88.

  5. R. Lidl and H. Niederreiter, Finite fields, Massachusetts, Addison-Wesley, Reading, 1983.
    MathSciNet    

  6. D. W. Masser and J. H. Rickert, Simultaneous Pell equations, J. Number Theory 61 (1996), 52-66.
    MathSciNet     CrossRef

  7. K. Ono, Euler's concordant forms, Acta Arith. 78 (1996), 101-123.
    MathSciNet    

  8. J. H. Rickert, Simultaneous rational approximations and related diophantine equations, Math. Proc. Cambridge Philos. Soc. 113 (1993), 461-472.
    MathSciNet     CrossRef

  9. H. P. Schlickewei, The number of subspaces occuring in the p-adic subspace theorem in diophantine approximation, J. Reine Angew. Math. 406 (1990), 44-108.
    MathSciNet     CrossRef

  10. W. M. Schmidt, Diophantine approximation, Springer Verlag, New York, 1980.
    MathSciNet    

  11. C. L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss. 1 (1929), 1-70.

  12. A. Thue, Über Annäherungswerte algebraischer Zahlen, J. Reine Angew. Math. 135 (1909), 284-305.

  13. P.-Z. Yuan, On the number of solutions of simultaneous Pell equations, Acta Arith. 101 (2002), 215-221.
    MathSciNet     CrossRef

  14. P.-Z. Yuan, Simultaneous Pell equations, Acta Arith. 115 (2004), 119-131.
    MathSciNet     CrossRef

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