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://www.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://www.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://www.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://www.numdam.org/articles/10.5802/jtnb.757/
[1] A. Balog and A. Sárközy, On sums of integers having small prime factors. I. Stud. Sci. Math. Hungar. 19 (1984), 35–47. | MR | Zbl
[2] A. Balog and A. Sárközy , On sums of integers having small prime factors. II. Stud. Sci. Math. Hungar. 19 (1984), 81–88. | MR | Zbl
[3] E. Bombieri and W. Gubler, Heights in Diophantine Geometry. Cambridge University Press, Cambridge, 2006. | MR | Zbl
[4] R. de la Bretèche, Sommes d’exponentielles et entiers sans grand facteur premier. Proc. London Math. Soc. 77 (1998), 39–78. | MR | Zbl
[5] R. de la Bretèche, Sommes sans grand facteur premier. Acta Arith. 88 (1999), 1–14. | EuDML | MR | Zbl
[6] R. de la Bretèche and A. Granville, Densité des friables. Preprint (2009).
[7] R. de la Bretèche and G. Tenenbaum, Séries trigonométriques à coefficients arithmétiques. J. Anal. Math. 92 (2004), 1–79. | MR | Zbl
[8] R. de la Bretèche and G. Tenenbaum, Propriétés statistiques des entiers friables, Ramanujan J. 9 (2005), 139–202. | MR | Zbl
[9] R. de la Bretèche and G. Tenenbaum, Sommes d’exponentielles friables d’arguments rationnels, Funct. Approx. Comment. Math. 37 (2007), 31–38. | MR | Zbl
[10] H. Davenport, Multiplicative Number Theory. Second Edition (Revised by H. L. Montgomery). Springer-Verlag, New York, 1980. | MR | Zbl
[11] P. Erdős, C. Stewart and R. Tijdeman, Some diophantine equations with many solutions. Compositio Math. 66 (1988), 37–56. | Numdam | MR | Zbl
[12] A. Granville and H. M. Stark, implies no “Siegel zeros” for -functions of characters with negative discriminant. Invent. Math. 139 (2000), 509–523. | MR | Zbl
[13] B. Gross and D. B. Zagier, On singular moduli. J. reine Angew. Math. 355 (1985), 191–220. | MR | Zbl
[14] K. Győry, On the Conjecture in algebraic number fields. Acta Arith. 133, (2008), no. 3, 283–295. | MR | Zbl
[15] K. Győry and K. Yu, Bounds for the solutions of -unit equations and decomposable form equations. Acta Arith. 123 (2006), no. 1, 9–41. | MR | Zbl
[16] G. H. Hardy and J. E. Littlewood, Some problems in partitio numerorum III. On the expression of a number as a sum of primes. Acta Math. 44 (1923), 1–70. | MR
[17] G.H. Hardy and J. E. Littlewood, Some problems in partitio numerorum V. A further contribution to the study of Goldbach’s problem. Proc. London Math. Soc., Ser. 2, 22 (1924), 46–56.
[18] A. Hildebrand, Integers free of large prime factors and the Riemann hypothesis. Mathematika 31 (1984), 258–271. | MR | Zbl
[19] A. Hildebrand, On the local behavior of . Trans. Amer. Math. Soc. 297 (1986), 729–751. | MR | Zbl
[20] A. Hildebrand and G. Tenenbaum, On integers free of large prime factors. Trans. Amer. Math. Soc. 296 (1986), 265–290. | MR | Zbl
[21] A. Hildebrand and G. Tenebaum, Integers without large prime factors. J. Theor. Nombres Bordeaux 5 (1993), 411–484. | Numdam | MR | Zbl
[22] S. Konyagin and K. Soundararajan, Two -unit equations with many solutions. J. Number Theory 124 (2007), 193–199. | MR | Zbl
[23] J. C. Lagarias and K. Soundararajan, Counting smooth solutions to the equation . Proc. London Math. Soc., to appear.
[24] D. W. Masser, On and discriminants. Proc. Amer. Math. Soc. 130 (2002), 3141–3150. | MR | Zbl
[25] J. Oesterlé, Nouvelles approches du “théorème” de Fermat. Sém. Bourbaki, Exp. No. 694, Astérisque No. 161-162 (1988), 165–186 (1989). | Numdam | MR | Zbl
[26] B. Poonen, E. F. Schaefer and M. Stoll, Twists of and primitive solutions to . Duke Math. J. 137 (2007), 103–158. | MR | Zbl
[27] C. L. Stewart and Kunrui Yu, On the abc Conjecture II. Duke Math. J. 108 (2001), 169–181. | MR | Zbl
[28] C. L. Stewart and R. Tijdeman, On the Oesterlé-Masser Conjecture. Monatshefte Math. 102 (1986), 251–257. | MR | Zbl
[29] G. Tenenbaum, Introduction to analytic and probabilistic number theory. Cambridge Univ. Press, Cambridge, 1995. | MR | Zbl
[30] R. C. Vaughan, The Hardy-Littlewood Method, Second Edition. Cambridge Tracts in Mathematics 125, Cambridge Univ. Press, 1997. | MR | Zbl
[31] B. M. M. de Weger, Solving exponential Diophantine equations using lattice basis reduction algorithms. J. Number Theory 26 (1987), no. 3, 325–367. | MR | Zbl
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