Abstract
In the present paper we give a survey of the abc-conjecture and of its modifications and generalizations. We discuss several consequences of the conjecture. At the end of the paper there are given numerical examples giving some evidence for the conjecture.
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References
N. Broberg, Some examples related to the abc-conjecture for algebraic number fields, Math. Comp. (to appear).
J. Browkin and Brzeziński, J., Some remarks on the abc-conjecture, Math. Comp. 62 (1994), no. 206, 931–939.
J. Browkin, Filaseta, M., Greaves, G. and Schinzel, A., Squarefree values of polynomials and the abc-conjecture, in: Sieve methods, exponential sums, and their applications in number theory ( G.R.H. Greaves, G. Harman, M.N. Huxley, eds.), Cambridge University Press, Cambridge 1996, pp. 65–85.
W.D. Brownawell and Masser, D.W., Vanishing sums in function fields, Math. Proc. Cambridge Philos. Soc. 100 (1986), 427–434.
L.V. Danilov, Letter to the editors, Mat. Zam. 36 (1984), 457–459. (Russian)
D. Davies, A note on the limit points associated with the generalized abc-conjecture for Z[t], Coll. Math. 71 (1996), 329–333.
N.D. Elkies, ABC implies Mordell, Internat. Math. Res. Notices 7 (1991), 99–109; in: Duke Math. Journ. 64 (1991).
M. Filaseta and Konyagin, S., On a limit point associated with the abc-conjecture. Coll. Math. 76 (1998), 265–268.
D. Goldfeld and Szpiro, L., Bounds for the order of the Tate-Shafarevich group, Compositio Math. 97 (1995), 71–87.
A. Granville, ABC allows us to count squarefrees,to appear.
A. Granville and Stark, H.M., ABC implies no ‘Siegel zero’,preprint.
G. Greaves and Nitaj, A., Some polynomial identities related to the abc-conjecture, in: Number Theory in Progress ( K. Györy, H. Iwaniec, J. Urbanowicz, eds.), W. de Gruyter, Berlin, 1999, pp. 229–236.
B.H. Gross and Zagier, D.B., Heegner points and derivatives of L-series, Invent. Math. 84 (1986), 225–320.
M. Hall, Jr., The diophantine equation x 3 − y 2 = k, Computers in Number Theory (A.O.L. Atkin, B.J. Birch, eds.), Academic Press, London 1971, pp. 173–198.
J. Kanapka, private communication.
S. Konyagin, private communication.
S. Lang, Old and new conjectured diophantine inequalities, Bull. Amer. Math. Soc. 23 (1990), 37–75.
R.C. Mason, Equations over function fields, Lecture Notes in Math. 1068 (1984), 149–157.
D.W. Masser, Note on a conjecture of Szpiro, Astérisque 183 (1990), 19–23.
A. Nitaj, La conjecture abc, L’Enseign. Math. 42 (1996), 3–24.
A. Nitaj, Invariants des courbes de Frey-Hellegouarch et grands groupes de TateShafarevich, preprint, 1998.
J. Oesterlé, Nouvelles approaches du “théorème” de Fermat., Sém. Bourbaki 1987–1988, Astérisque 161–162 (1988), no. 694, 165–186.
M. Overholt, The diophantine equation n! + 1 = m 2, Bull. London Math. Soc. 25 (1993), 104.
K. Prachar, Primzahlverteilung, Springer, Berlin, 1957.
H.N. Shapiro and Sparer, G.H., Extension of a theorem of Mason, Comm. Pure and Appl. Math. 47 (1994), 711–718.
W. Sierpifiski, Elementary Theory of Numbers, Elsevier, 1988, p. 105.
J.H. Silverman, Wieferich’s criterion and the abc-conjecture, J. Number Theory 30 (1988), 226–237.
C.L. Stewart and Tijdeman, R., On the Oesterlé-Masser conjecture, Monatsh. Math. 102 (1986), 251–257.
C.L. Stewart and Kunrui Yu, On the abc-conjecture, Math. Ann. 291 (1991), 225–230.
C.L. Stewart and Kunrui Yu, Lecture given by C.L. Stewart during the Conference in Zakopane, (1997).
W.W. Stothers, Polynomial identities and Hauptmoduln, Quart. J. Math. Oxford (2), 32 (1981), 349–370.
L. Szpiro, Discriminant et conducteur, Astérisque 183 (1990) 7–17.
P. Vojta, Diophantine approximation and value distribution theory, Lecture Notes in Math. vol. 1239, Springer-Verlag, 1987.
J.F. Voloch, Diagonal equations over function fields, Bol. Soc. Brasil. Mat. 16 (1985), 29–39.
B.M.M. de Weger, Algorithms for diophantine equations, CWI Tract, Centr. Math. Comput. Sci., Amsterdam, 1989.
B.M.M. de Weger, A + B = C and big Ш’s, Quart. J. Math. Oxford (2), 49 (1998), no. 193, 105–128.
U. Zannier, Some remarks on the S-unit equation in function fields, Acta Arith. 64 (1993), 87–98
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Browkin, J. (2000). The abc-conjecture. In: Bambah, R.P., Dumir, V.C., Hans-Gill, R.J. (eds) Number Theory. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7023-8_5
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