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(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.

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.

Published online:
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/

[1] A. Balog and A. Sárközy, On sums of integers having small prime factors. I. Stud. Sci. Math. Hungar. 19 (1984), 35–47. | MR: 787784 | Zbl: 0569.10025

[2] A. Balog and A. Sárközy , On sums of integers having small prime factors. II. Stud. Sci. Math. Hungar. 19 (1984), 81–88. | MR: 787784 | Zbl: 0569.10026

[3] E. Bombieri and W. Gubler, Heights in Diophantine Geometry. Cambridge University Press, Cambridge, 2006. | MR: 2216774 | Zbl: 1130.11034

[4] R. de la Bretèche, Sommes d’exponentielles et entiers sans grand facteur premier. Proc. London Math. Soc. 77 (1998), 39–78. | MR: 1625487 | Zbl: 0893.11039

[5] R. de la Bretèche, Sommes sans grand facteur premier. Acta Arith. 88 (1999), 1–14. | EuDML: 207229 | MR: 1698349 | Zbl: 0935.11031

[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: 2072741 | Zbl: 1171.11319

[8] R. de la Bretèche and G. Tenenbaum, Propriétés statistiques des entiers friables, Ramanujan J. 9 (2005), 139–202. | MR: 2166385 | Zbl: 1152.11040

[9] R. de la Bretèche and G. Tenenbaum, Sommes d’exponentielles friables d’arguments rationnels, Funct. Approx. Comment. Math. 37 (2007), 31–38. | MR: 2357307 | Zbl: pre05257413

[10] H. Davenport, Multiplicative Number Theory. Second Edition (Revised by H. L. Montgomery). Springer-Verlag, New York, 1980. | MR: 606931 | Zbl: 0453.10002

[11] P. Erdős, C. Stewart and R. Tijdeman, Some diophantine equations with many solutions. Compositio Math. 66 (1988), 37–56. | Numdam | MR: 937987 | Zbl: 0639.10014

[12] A. Granville and H. M. Stark, abc implies no “Siegel zeros” for L-functions of characters with negative discriminant. Invent. Math. 139 (2000), 509–523. | MR: 1738058 | Zbl: 0967.11033

[13] B. Gross and D. B. Zagier, On singular moduli. J. reine Angew. Math. 355 (1985), 191–220. | MR: 772491 | Zbl: 0545.10015

[14] K. Győry, On the abc Conjecture in algebraic number fields. Acta Arith. 133, (2008), no. 3, 283–295. | MR: 2434605 | Zbl: 1188.11011

[15] K. Győry and K. Yu, Bounds for the solutions of S-unit equations and decomposable form equations. Acta Arith. 123 (2006), no. 1, 9–41. | MR: 2232500 | Zbl: 1163.11026

[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: 1555183

[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: 804201 | Zbl: 0544.10042

[19] A. Hildebrand, On the local behavior of Ψ(x,y). Trans. Amer. Math. Soc. 297 (1986), 729–751. | MR: 854096 | Zbl: 0611.10028

[20] A. Hildebrand and G. Tenenbaum, On integers free of large prime factors. Trans. Amer. Math. Soc. 296 (1986), 265–290. | MR: 837811 | Zbl: 0601.10028

[21] A. Hildebrand and G. Tenebaum, Integers without large prime factors. J. Theor. Nombres Bordeaux 5 (1993), 411–484. | Numdam | MR: 1265913 | Zbl: 0797.11070

[22] S. Konyagin and K. Soundararajan, Two S-unit equations with many solutions. J. Number Theory 124 (2007), 193–199. | MR: 2321000 | Zbl: 1131.11022

[23] J. C. Lagarias and K. Soundararajan, Counting smooth solutions to the equation A+B=C. Proc. London Math. Soc., to appear.

[24] D. W. Masser, On abc and discriminants. Proc. Amer. Math. Soc. 130 (2002), 3141–3150. | MR: 1912990 | Zbl: 1030.11011

[25] J. Oesterlé, Nouvelles approches du “théorème” de Fermat. Sém. Bourbaki, Exp. No. 694, Astérisque No. 161-162 (1988), 165–186 (1989). | Numdam | MR: 992208 | Zbl: 0668.10024

[26] B. Poonen, E. F. Schaefer and M. Stoll, Twists of X(7) and primitive solutions to x 2 +y 3 =z 7 . Duke Math. J. 137 (2007), 103–158. | MR: 2309145 | Zbl: 1124.11019

[27] C. L. Stewart and Kunrui Yu, On the abc Conjecture II. Duke Math. J. 108 (2001), 169–181. | MR: 1831823 | Zbl: 1036.11032

[28] C. L. Stewart and R. Tijdeman, On the Oesterlé-Masser Conjecture. Monatshefte Math. 102 (1986), 251–257. | MR: 863221 | Zbl: 0597.10042

[29] G. Tenenbaum, Introduction to analytic and probabilistic number theory. Cambridge Univ. Press, Cambridge, 1995. | MR: 1342300 | Zbl: 0831.11001

[30] R. C. Vaughan, The Hardy-Littlewood Method, Second Edition. Cambridge Tracts in Mathematics 125, Cambridge Univ. Press, 1997. | MR: 1435742 | Zbl: 0868.11046

[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: 901244 | Zbl: 0625.10013

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