Smooth solutions to the $abc$ equation: the $xyz$ Conjecture
Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 1, pp. 209-234.

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 $H\left(A,B,C\right)=max\left(|A|,|B|,|C|\right)$, and consider primitive solutions ($\mathrm{gcd}\left(A,B,C\right)=1$) having no prime factor larger than ${\left(logH\left(A,B,C\right)\right)}^{\kappa }$, 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.

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 $H\left(A,B,C\right)=max\left(|A|,|B|,|C|\right)$, et on considère les solutions primitives ($gcd\left(A,B,C\right)=1$) n’ayant aucun facteur premier plus grand que ${\left(logH\left(A,B,C\right)\right)}^{\kappa }$, 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.

DOI: 10.5802/jtnb.757
Jeffrey C. Lagarias 1; Kannan Soundararajan 2

1 University of Michigan Department of Mathematics 530 Church Street Ann Arbor, MI 48109-1043, USA
2 Department of Mathematics Stanford University Department of Mathematics Stanford, CA 94305-2025,USA
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Jeffrey C. Lagarias; Kannan Soundararajan. Smooth solutions to the $abc$ equation: the $xyz$ Conjecture. Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 1, pp. 209-234. doi : 10.5802/jtnb.757. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.757/

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