abc conjecture

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English

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Alternative forms

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Etymology

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From the names of the variables a, b, and c. The conjecture was proposed in 1985.

Noun

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abc conjecture (plural abc conjectures)

  1. (number theory) Given coprime positive integers a, b and c, such that a + b = c, and d the radical of abc (the product of its distinct prime factors), the conjecture that d is usually not much smaller than c (in other words, that if a and b are divisible by large powers of primes, then c usually is not).
    • 1985, Paul Vojta, Appendix, Serge Lang, Introduction to Arakelov Theory, Springer, 1988 Softcover, page 156,
      Finally in §5 we give one application to the curve X4 + Y4 = Z4, showing that the height inequalities for the curve imply the asymptotic Fermat conjecture and a weak form of the Masser-Oesterlé abc conjecture.
    • 2004, Sergei K. Lando, R.V. Gamkrelidze, V.A. Vassiliev, Graphs on Surfaces and Their Applications, Springer, page 137:
      The abc conjecture may well replace the Fermat theorem for the future generation of mathematicians.
    • 2006, Pei-Chu Hu, Chung-Chun Yang, Value Distribution Theory Related to Number Theory, Springer (Birkhäuser), page 233:
      To prove or disprove the abc-conjecture would be an important contribution to number theory. [] Langevin ([236], [237]) proved that the abc-conjecture implies the Erdős-Woods conjecture with k = 3 except perhaps a finite number of counter examples.
    • 2007, Enrico Bombieri, Walter Gubler, Heights in Diophantine Geometry, Cambridge University Press, page 401:
      The abc-conjecture of Masser and Oesterle is a typical example of a simple statement that can be used to unify and motivate many results in number theory, which otherwise would be scattered statements without a common link.
  2. Any of certain generalisations of the conjecture.

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Further reading

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