Smooth solutions to the abc equation: the xyz Conjecture
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(|A|,|B|,|C|), et on considère les solutions primitives (gcd(A,B,C)=1) n’ayant aucun facteur premier plus grand que (logH(A,B,C)) κ , pour un κ fini donné. Nous montrons que la Conjecture abc entraine que pour tout κ<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 κ>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(|A|,|B|,|C|), and consider primitive solutions ( gcd (A,B,C)=1) having no prime factor larger than (logH(A,B,C)) κ , for a given finite κ. We show that the abc Conjecture implies that for any fixed κ<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 κ>8 the abc equation has infinitely many primitive solutions. We outline a proof of the latter result.

@article{JTNB_2011__23_1_209_0,
     author = {Jeffrey C. Lagarias and Kannan Soundararajan},
     title = {Smooth solutions to the $abc$ equation: the $xyz$ {Conjecture}},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {209--234},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {23},
     number = {1},
     year = {2011},
     doi = {10.5802/jtnb.757},
     zbl = {1270.11032},
     mrnumber = {2780626},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.757/}
}
Jeffrey C. Lagarias; Kannan Soundararajan. Smooth solutions to the $abc$ equation: the $xyz$ Conjecture. Journal de Théorie des Nombres de Bordeaux, Tome 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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