Why Quadratic Diophantine Equations?

  • Titu Andreescu
  • Dorin Andrica
Part of the Developments in Mathematics book series (DEVM, volume 40)


In order to motivate the study of quadratic type equations, in this chapter we present several problems from various mathematical disciplines leading to such equations. The diversity of the arguments to follow underlines the importance of this subject.


Modular Form Diophantine Equation Fuchsian Group Torsion Subgroup Binary Quadratic Form 
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  1. 59.
    Davis, M., Matiyasevich, Yu., Robinson, J.: Diophantine equations: a positive aspect of a negative solution. Mathematical developments arising from Hilbert problems. Proc. Symp. Pure Math. 28, 323–378 (1976). Am. Math. Soc., ProvidenceGoogle Scholar
  2. 62.
    Dickson, L.E.: Introduction to the theory of numbers. Dover, New York (1957)MATHGoogle Scholar
  3. 94.
    Hua, L.K., Wang, Y.: On Diophantine approximations and numerical integrations I, II. Sci. Sinica 13, 1007–1009 (1964) and 13, 1009–1010 (1964)Google Scholar
  4. 110.
    Lehner, J., Sheingorn, M.: Computing self-intersections of closed geodesics on finite-sheeted covers of the modular surface. Math. Comp. 44(169), 233–240 (1985)MATHMathSciNetCrossRefGoogle Scholar
  5. 113.
    Li, X.-J.: On the Trace of Hecke Operators for Maass Forms. Number Theory (Ottawa, ON, 1996). CRM Proc. Lecture Notes, vol. 19, pp. 215–229. Am. Math. Soc., Providence (1999)Google Scholar
  6. 124.
    Manin, Yu.I., Panchishkin, A.A.: Introduction to Modern Number Theory, 2nd edn. Springer, New York (2005)MATHCrossRefGoogle Scholar
  7. 130.
    Matiyasevich, Yu.: Le dixième problème de Hilbert: que peut-on faire avec les équations diophantiennes? La Recherche de la Vérité, coll. L’écriture des Mathématiques, ACL - Les Éditions du Kangourou, pp. 281–305 (1999)Google Scholar
  8. 131.
    Matiyasevich, Yu.: Hilbert’s Tenth Problem: Diophantine Equations from Algorithmic Point of View. Hilbert’s Problems Today, 5th–7th April 2001. Pisa, Italy (2001)Google Scholar
  9. 132.
    Matiyasevich, Yu.: Hilbert’s Tenth Problem. MIT, Cambridge (1993)Google Scholar
  10. 156.
    Nikonorov, Y.G., Rodionov, E.D.: Standard homogeneous Einstein manifolds and Diophantine equations. Arch. Math. (Brno) 32(2), 123–136 (1996)Google Scholar
  11. 166.
    Ono, K.: Euler’s concordant forms. Acta Arith. 78(2), 101–126 (1996)MathSciNetGoogle Scholar
  12. 186.
    Rosen, D., Schmidt, T.A.: Hecke groups and continued fractions. Bull. Aust. Math. Soc. 46(3), 459–474 (1992)MathSciNetCrossRefGoogle Scholar
  13. 197.
    Seidenberg, A.: A new decision method for elementary algebra. Ann. Math. 60(1954), 365–374MATHMathSciNetCrossRefGoogle Scholar
  14. 209.
    Tallini, G.: Some new results on sets of type (m, n) in projective planes. J. Geom. 29(2), 191–199 (1987)MATHMathSciNetCrossRefGoogle Scholar
  15. 211.
    Tarski, A.: A Decision Method for Elementary Algebra and Geometry. University of California Press, Berkeley (1951)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Titu Andreescu
    • 1
  • Dorin Andrica
    • 2
  1. 1.School of Natural Sciences and MathematicsUniversity of Texas at DallasRichardsonUSA
  2. 2.Faculty of Mathematics and Computer Science“Babeş-Bolyai” UniversityCluj-NapocaRomania

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