abc conjecture

English

Alternative forms

• abc-conjecture
• ABC conjecture

Etymology

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

Noun

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.

Synonyms

• (conjecture in number theory): Oesterlé-Masser conjecture

Translations

conjecture in number theory

• French: conjecture abc f
• German: abc-Vermutung f
• Italian: congettura abc f
• Portuguese: conjetura abc f
• Spanish: conjetura abc f

Further reading

• Radical of an integer on Wikipedia.Wikipedia