This paper studies integer solutions to the equation in which none of have a large prime factor. We set , and consider primitive solutions () having no prime factor larger than , for a given finite . We show that the Conjecture implies that for any fixed 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 the equation has infinitely many primitive solutions. We outline a proof of the latter result.
Cet article étudie les solutions entières de l’équation pour lesquelles ni , ni , ni n’ont de grands facteurs premiers. On pose , et on considère les solutions primitives () n’ayant aucun facteur premier plus grand que , pour un fini donné. Nous montrons que la Conjecture entraine que pour tout 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 l’équation a un nombre infini de solutions primitives. Nous esquissons la preuve de ce dernier résultat.
@article{JTNB_2011__23_1_209_0, author = {Lagarias, Jeffrey C. and Soundararajan, Kannan}, 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 = {http://archive.numdam.org/articles/10.5802/jtnb.757/} }
TY - JOUR AU - Lagarias, Jeffrey C. AU - Soundararajan, Kannan TI - Smooth solutions to the $abc$ equation: the $xyz$ Conjecture JO - Journal de théorie des nombres de Bordeaux PY - 2011 SP - 209 EP - 234 VL - 23 IS - 1 PB - Société Arithmétique de Bordeaux UR - http://archive.numdam.org/articles/10.5802/jtnb.757/ DO - 10.5802/jtnb.757 LA - en ID - JTNB_2011__23_1_209_0 ER -
%0 Journal Article %A Lagarias, Jeffrey C. %A Soundararajan, Kannan %T Smooth solutions to the $abc$ equation: the $xyz$ Conjecture %J Journal de théorie des nombres de Bordeaux %D 2011 %P 209-234 %V 23 %N 1 %I Société Arithmétique de Bordeaux %U http://archive.numdam.org/articles/10.5802/jtnb.757/ %R 10.5802/jtnb.757 %G en %F JTNB_2011__23_1_209_0
Lagarias, Jeffrey C.; Soundararajan, Kannan. 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. http://archive.numdam.org/articles/10.5802/jtnb.757/
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