The Diophantine equation x 2−(t 2+t)y 2−(4t+2)x+(4t 2+4t)y=0

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Abstract

Let t≥1 be an integer. In this work, we consider the number of integer solutions of Diophantine equation
$$x^{2}-(t^{2}+t)y^{2}-(4t+2)x+(4t^{2}+4t)y=0$$
over ℤ and also over finite fields \(\mathbb{F}_{p}\) for primes p≥5.
Diophantine equation Pell equation 

Mathematics Subject Classification (2000)

11D09 11D41 11D45 

References

  1. 1.
    Arya, S.P.: On the Brahmagupta-Bhaskara equation. Math. Educ. 8(1), 23–27 (1991) MathSciNetGoogle Scholar
  2. 2.
    Baltus, C.: Continued fractions and the Pell equations: the work of Euler and Lagrange. Comm. Anal. Theory Contin. Fractions 3, 4–31 (1994) MathSciNetGoogle Scholar
  3. 3.
    Barbeau, E.: Pell’s Equation. Springer, Berlin (2003) MATHGoogle Scholar
  4. 4.
    Edwards, H.M.: Fermat’s Last Theorem. A Genetic Introduction to Algebraic Number Theory. Graduate Texts in Mathematics, vol. 50. Springer, New York (1996). Corrected reprint of the 1977 original MATHGoogle Scholar
  5. 5.
    Hensley, D.: Continued Fractions. World Scientific, Singapore (2006) MATHGoogle Scholar
  6. 6.
    Kaplan, P., Williams, K.S.: Pell’s equations x 2my 2=−1,−4 and continued fractions. J. Number Theory 23, 169–182 (1986) CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Lenstra, H.W.: Solving the Pell equation. Not. AMS 49(2), 182–192 (2002) MathSciNetMATHGoogle Scholar
  8. 8.
    Matthews, K.: The Diophantine equation x 2Dy 2=N,D>0. Expo. Math. 18, 323–331 (2000) MathSciNetMATHGoogle Scholar
  9. 9.
    Mollin, R.A., Poorten, A.J., Williams, H.C.: Halfway to a solution of x 2Dy 2=−3. J. Theor. Nr. Bordx. 6, 421–457 (1994) MATHGoogle Scholar
  10. 10.
    Mollin, R.A.: Simple continued fraction solutions for Diophantine equations. Expo. Math. 19(1), 55–73 (2001) MathSciNetMATHGoogle Scholar
  11. 11.
    Mollin, R.A., Cheng, K., Goddard, B.: The Diophantine equation AX 2BY 2=C solved via continued fractions. Acta Math. Univ. Comen. 71(2), 121–138 (2002) MathSciNetMATHGoogle Scholar
  12. 12.
    Mollin, R.A.: The Diophantine equation ax 2by 2=c and simple continued fractions. Int. Math. J. 2(1), 1–6 (2002) MathSciNetMATHGoogle Scholar
  13. 13.
    Mollin, R.A.: A continued fraction approach to the Diophantine equation ax 2by 2=±1. JP J. Algebra Number Theory Appl. 4(1), 159–207 (2004) MathSciNetMATHGoogle Scholar
  14. 14.
    Mollin, R.A.: A note on the Diophantine equation D 1 x 2+D 2=ak n. Acta Math. Acad. Paedagog. Nyireg. 21, 21–24 (2005) MathSciNetMATHGoogle Scholar
  15. 15.
    Mollin, R.A.: Quadratic Diophantine equations x 2Dy 2=c n. Ir. Math. Soc. Bull. 58, 55–68 (2006) MathSciNetMATHGoogle Scholar
  16. 16.
    Niven, I., Zuckerman, H.S., Montgomery, H.L.: An Introduction to the Theory of Numbers, 5th edn. Wiley, New York (1991) Google Scholar
  17. 17.
    Stevenhagen, P.: A density conjecture for the negative Pell equation. Comput. Algebra Number Theory Math. Appl. 325, 187–200 (1992) MathSciNetGoogle Scholar
  18. 18.
    Tekcan, A.: Pell equation x 2Dy 2=2, II. Bull. Ir. Math. Soc. 54, 73–89 (2004) MathSciNetMATHGoogle Scholar
  19. 19.
    Tekcan, A., Bizim, O., Bayraktar, M.: Solving the Pell equation using the fundamental element of the field \(\mathbb{Q}(\sqrt{\Delta})\). South East Asian Bull. Math. 30, 355–366 (2006) MathSciNetMATHGoogle Scholar
  20. 20.
    Tekcan, A.: The Pell equation x 2Dy 2=±4. Appl. Math. Sci. 1(8), 363–369 (2007) MathSciNetMATHGoogle Scholar
  21. 21.
    Tekcan, A., Gezer, B., Bizim, O.: On the integer solutions of the Pell equation x 2dy 2=2t. Int. J. Comput. Math. Sci. 1(3), 204–208 (2007) MathSciNetGoogle Scholar
  22. 22.
    Tekcan, A.: The Pell equation x 2−(k 2k)y 2=2t. Int. J. Comput. Math. Sci. 2(1), 5–9 (2008) CrossRefMathSciNetGoogle Scholar
  23. 23.
    Tekcan, A.: The Diophantine equation 4y 2−4yx−1=0, curves and conics over finite fields. Math. Rep. (accepted for publication) Google Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Faculty of Science, Department of MathematicsUludag UniversityGörükleTurkey

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