Sums of two 𝑆-units via Frey-Hellegouarch curves

Bennett, Michael; Billerey, Nicolas

doi:10.1090/mcom/3129

Sums of two $S$-units via Frey-Hellegouarch curves
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by Michael A. Bennett and Nicolas Billerey;

Math. Comp. 86 (2017), 1375-1401

DOI: https://doi.org/10.1090/mcom/3129

Published electronically: August 18, 2016

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Abstract:

In this paper, we develop a new method for finding all perfect powers which can be expressed as the sum of two rational $S$-units, where $S$ is a finite set of primes. Our approach is based upon the modularity of Galois representations and, for the most part, does not require lower bounds for linear forms in logarithms. Its main virtue is that it enables us to carry out such a program explicitly, at least for certain small sets of primes $S$; we do so for $S = \{ 2, 3 \}$ and $S= \{ 3, 5, 7 \}$.

References

Michael A. Bennett, Products of consecutive integers, Bull. London Math. Soc. 36 (2004), no. 5, 683–694. MR 2070445, DOI 10.1112/S0024609304003480
M. A. Bennett, K. Győry, M. Mignotte, and Á. Pintér, Binomial Thue equations and polynomial powers, Compos. Math. 142 (2006), no. 5, 1103–1121. MR 2264658, DOI 10.1112/S0010437X06002181
Michael A. Bennett and Chris M. Skinner, Ternary Diophantine equations via Galois representations and modular forms, Canad. J. Math. 56 (2004), no. 1, 23–54. MR 2031121, DOI 10.4153/CJM-2004-002-2
Michael A. Bennett, Vinayak Vatsal, and Soroosh Yazdani, Ternary Diophantine equations of signature $(p,p,3)$, Compos. Math. 140 (2004), no. 6, 1399–1416. MR 2098394, DOI 10.1112/S0010437X04000983
Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, On the modularity of elliptic curves over $\mathbf Q$: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939. MR 1839918, DOI 10.1090/S0894-0347-01-00370-8
Armand Brumer and Kenneth Kramer, The conductor of an abelian variety, Compositio Math. 92 (1994), no. 2, 227–248. MR 1283229
J. E. Cremona, Database of elliptic curves, 2014. SAGE package available at http://www.sagemath.org/packages/.
J. E. Cremona and M. P. Lingham, Finding all elliptic curves with good reduction outside a given set of primes, Experiment. Math. 16 (2007), no. 3, 303–312. MR 2367320
Sander R. Dahmen and Samir Siksek, Perfect powers expressible as sums of two fifth or seventh powers, Acta Arith. 164 (2014), no. 1, 65–100. MR 3223319, DOI 10.4064/aa164-1-5
Mou-Jie Deng, A note on the Diophantine equation $x^2+q^m=c^{2n}$, Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), no. 2, 15–18. MR 3310965, DOI 10.3792/pjaa.91.15
Clemens Fuchs, Rafael von Känel, and Gisbert Wüstholz, An effective Shafarevich theorem for elliptic curves, Acta Arith. 148 (2011), no. 2, 189–203. MR 2786163, DOI 10.4064/aa148-2-5
K. Hambrook, Implementation of a Thue-Mahler solver, M.Sc. thesis, University of British Columbia, 2011.
R. von Känel, Modularity and integral points on moduli schemes, preprint.
D. Kim, A modular approach to Thue-Mahler equations, preprint.
Alain Kraus, Majorations effectives pour l’équation de Fermat généralisée, Canad. J. Math. 49 (1997), no. 6, 1139–1161 (French, with French summary). MR 1611640, DOI 10.4153/CJM-1997-056-2
M. Ram Murty and Hector Pasten, Modular forms and effective Diophantine approximation, J. Number Theory 133 (2013), no. 11, 3739–3754. MR 3084298, DOI 10.1016/j.jnt.2013.05.006
The PARI Group, Bordeaux, PARI/GP version 2.7.1, 2014, available at http://pari. math.u-bordeaux.fr/.
K. A. Ribet, On modular representations of $\textrm {Gal}(\overline \textbf {Q}/\textbf {Q})$ arising from modular forms, Invent. Math. 100 (1990), no. 2, 431–476. MR 1047143, DOI 10.1007/BF01231195
T. N. Shorey and R. Tijdeman, Exponential Diophantine equations, Cambridge Tracts in Mathematics, vol. 87, Cambridge University Press, Cambridge, 1986. MR 891406, DOI 10.1017/CBO9780511566042
Joseph H. Silverman, The arithmetic of elliptic curves, 2nd ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR 2514094, DOI 10.1007/978-0-387-09494-6
W. A. Stein et al., Sage Mathematics Software (Version 5.13). The Sage Development Team, 2013. http://www.sagemath.org.
C. Störmer, Solution complète en nombres entiers de l’équation $m\arctan \frac 1x+n\arctan \frac 1y=k\frac {\pi }4$, Bull. Soc. Math. France 27 (1899), 160–170 (French). MR 1504340
Nobuhiro Terai, A note on the Diophantine equation $x^2+q^m=c^n$, Bull. Aust. Math. Soc. 90 (2014), no. 1, 20–27. MR 3227126, DOI 10.1017/S0004972713000981
B. M. M. de Weger, Algorithms for Diophantine equations, CWI Tract, vol. 65, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1989. MR 1026936
B. M. M. de Weger, The weighted sum of two $S$-units being a square, Indag. Math. (N.S.) 1 (1990), no. 2, 243–262. MR 1060828, DOI 10.1016/0019-3577(90)90007-A