Smooth solutions to the $abc$ equation: the $xyz$ Conjecture

Lagarias, Jeffrey C.; Soundararajan, Kannan

Journal de théorie des nombres de Bordeaux, Tome 23 (2011) no. 1, pp. 209-234.

Cet article étudie les solutions entières de l’équation $abc$ pour lesquelles ni $A$, ni $B$, ni $C$ n’ont de grands facteurs premiers. On pose $H(A,B,C)=max\left(\right|A|,|B|,|C\left|\right)$, et on considère les solutions primitives ($gcd(A,B,C)=1$) n’ayant aucun facteur premier plus grand que ${(logH(A,B,C))}^{\kappa }$, pour un $\kappa$ fini donné. Nous montrons que la Conjecture $abc$ entraine que pour tout $\kappa <1$ l’équation n’a qu’un nombre fini de solutions primitives. Nous donnons aussi un résultat conditionnel, affirmant que l’hypothèse de Riemann généralisée (GRH) implique que pour tout $\kappa >8$ l’équation $abc$ a un nombre infini de solutions primitives. Nous esquissons la preuve de ce dernier résultat.

This paper studies integer solutions to the $abc$ equation $A+B+C=0$ in which none of $A,B,C$ have a large prime factor. We set $H(A,B,C)=max\left(\right|A|,|B|,|C\left|\right)$, and consider primitive solutions ($\gcd (A,B,C)=1$) having no prime factor larger than ${(logH(A,B,C))}^{\kappa }$, for a given finite $\kappa$. We show that the $abc$ Conjecture implies that for any fixed $\kappa <1$ the equation has only finitely many primitive solutions. We also discuss a conditional result, showing that the Generalized Riemann hypothesis (GRH) implies that for any fixed $\kappa >8$ the $abc$ equation has infinitely many primitive solutions. We outline a proof of the latter result.

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