Number Theory: An Introduction to Mathematics

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584 XIII Connections with Number Theory


For congruent numbers, see Volume II, Chapter XVI of Dickson [23], Tunnell [57],
and Noda and Wada [43]. The survey articles of Goldfeld [30] and Oesterl ́e [44] deal
with Gauss’s class number problem.
References for earlier work on Fermat’s last theorem were given in Chapter III.
Ribet [47] and Cornellet al. [20] provide some preparation for the original papers of
Wiles [60] and Taylor and Wiles [56]. For theTW-conjecture, see also Darmon [22].
For Euler’s conjecture, see Elkies [25].


8 SelectedReferences


[1] G.E. Andrews, A polynomial identity which implies the Rogers-Ramanujan identities,
Scripta Math. 28 (1970), 297–305.
[2] G.E. Andrews, The theory of partitions, Addison-Wesley, Reading, Mass., 1976.
[Paperback edition, Cambridge University Press, 1998]
[3] G.E. Andrews, R.A. Askey, B.C. Berndt, K.G. Ramanathan and R.A. Rankin (ed.),
Ramanujan revisited, Academic Press, London, 1988.
[4] G.E. Andrews, R. Askey and R. Roy,Special functions, Cambridge University Press, 1999.
[5] L. B ́aez-Duarte, Hardy-Ramanujan’s asymptotic formula for partitions and the central
limit theorem,Adv. in Math. 125 (1997), 114–120.
[6] A. Baker, The diophantine equationy^2 =ax^3 +bx^2 +cx+d,J. London Math. Soc. 43
(1968), 1–9.
[7] A. Baker, The theory of linear forms in logarithms,Transcendence theory:advances and
applications(ed. A. Baker and D.W. Masser), pp. 1–27, Academic Press, London, 1977.
[8] R.J. Baxter,Exactly solved models in statistical mechanics, Academic Press, London,


  1. [Reprinted, 1989]
    [9] A. Berkovich and B.M. McCoy, Rogers-Ramanujan identities: a century of progress from
    mathematics to physics,Proceedings of the International Congress of Mathematicians:
    Berlin1998, Vol. III, pp. 163–172, Documenta Mathematica, Bielefeld, 1998.
    [10] J.S. Birman, New points of view in knot theory,Bull. Amer. Math. Soc.(N.S.) 28 (1993),
    253–287.
    [11] S. Bloch, A note on height pairings, Tamagawa numbers, and the Birch and Swinnerton-
    Dyer conjecture,Invent. Math. 58 (1980), 65–76.
    [12] S. Bloch, The proof of the Mordell conjecture,Math. Intelligencer 6 (1984), no. 2, 41–47.
    [13] D.M. Bressoud and D. Zeilberger, A short Rogers-Ramanujan bijection,Discrete Math.
    38 (1982), 313–315.
    [14] D.M. Bressoud and D. Zeilberger, Bijecting Euler’s partitions-recurrence,Amer. Math.
    Monthly 92 (1985), 54–55.
    [15] Y. Bugeaud, On the size of integer solutions of elliptic equations,Bull. Austral. Math. Soc.
    57 (1998), 199–206.
    [16] J.W.S. Cassels, Diophantine equationswith special reference to elliptic curves,J. London
    Math. Soc. 41 (1966), 193–291.
    [17] J.S. Chahal, Manin’s proof of the Hasse inequality revisited,Nieuw Arch. Wisk.(4) 13
    (1995), 219–232.
    [18] J.Ci ̆ ̆zm ́ar, Birationale Transformationen (Ein historischerUberblick), ̈ Per i od. Pol yt ech.
    Mech. Engrg. 39 (1995), 9–24.
    [19] G. Cornell and J.H. Silverman (ed.),Arithmetic geometry, Springer-Verlag, New York,


  2. [20] G. Cornell, J.H. Silverman and G. Stevens (ed.),Modular forms and Fermat’s last
    theorem, Springer, New York, 1997.



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