abc conjecture

Definitions

1. 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).
2. Any of certain generalisations of the conjecture.

Equivalents

Examples

“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.”

“The abc''' conjecture may well replace⟳ the Fermat theorem for the future generation of mathematicians.”

“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.”

“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⟳.”

CEFR level