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Lecture on the abc Conjecture and Some of Its Consequences

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Book cover Mathematics in the 21st Century

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 98))

Abstract

The abc conjecture was proposed in the 1980s by J. Oesterlé and D.W. Masser. This simple statement implies a number of results and conjectures in number theory. We state this conjecture and list a few of the many consequences. This conjecture has gained increasing awareness in August 2012 when Shinichi Mochizuki released a series of four preprints containing a claim to a proof of the abc conjecture using his inter-universal Teichmüller theory: http://www.kurims.kyoto-u.ac.jp/~motizuki/top-english.html

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Notes

  1. 1.

    See the tables of http://rekenmeemetabc.nl/Synthese_resultaten

  2. 2.

    Every integer is a cube or the sum of two, three, …nine cubes; every integer is also the square of a square, or the sum of up to nineteen such; and so forth. Similar laws may be affirmed for the correspondingly defined numbers of quantities of any like degree.

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  1. Université Pierre et Marie Curie-Paris 6, Institut de Mathématiques de Jussieu IMJ UMR 7586 Théorie des Nombres Case Courrier 247, 4 Place Jussieu, F–75252, Paris Cedex 05, France

    Michel Waldschmidt

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  1. Institut des Hautes Études Scientifiques (IHÉS), Bures-sur-Yvette, France

    Pierre Cartier

  2. Abdus Salam School of Mathematical Sciences, Lahore, Pakistan

    A.D.R. Choudary

  3. Université Pierre et Marie Curie (Paris VI),, Paris Cedex 05, France

    Michel Waldschmidt

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Waldschmidt, M. (2015). Lecture on the abc Conjecture and Some of Its Consequences. In: Cartier, P., Choudary, A., Waldschmidt, M. (eds) Mathematics in the 21st Century. Springer Proceedings in Mathematics & Statistics, vol 98. Springer, Basel. https://doi.org/10.1007/978-3-0348-0859-0_13

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