Shifted powers in Lucas–Lehmer sequences
Tóm tắt
We develop a general framework for finding all perfect powers in sequences derived via shifting non-degenerate quadratic Lucas–Lehmer binary recurrence sequences by a fixed integer. By combining this setup with bounds for linear forms in logarithms and results based upon the modularity of elliptic curves defined over totally real fields, we are able to answer a question of Bugeaud, Luca, Mignotte and the third author by explicitly finding all perfect powers of the shape
$$F_k \pm 2 $$
where
$$F_k$$
is the k-th term in the Fibonacci sequence.
Tài liệu tham khảo
Bennett, M.A., Dahmen, S., Mignotte, M., Siksek, S.: Shifted powers in binary recurrence sequences. Math. Proc. Camb. Philos. Soc. 158, 305–329 (2015)
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: the user language, J. Symb. Comp. 24, 235–265. (See also http://magma.maths.usyd.edu.au/magma/) (1997)
Bugeaud, Y., Luca, F., Mignotte, M., Siksek, S.: Fibonacci numbers at most one away from a perfect power. Elem. Math. 63(2), 65–75 (2008)
Bugeaud, Y., Luca, F., Mignotte, M., Siksek, S.: Almost powers in the Lucas sequences. J. Théor. Nombres Bordeaux 20, 555–600 (2008)
Bugeaud, Y., Mignotte, M., Siksek, S.: Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers. Ann. Math 163, 969–1018 (2006)
Bugeaud, Y., Mignotte, M., Siksek, S., Stoll, M., Tengely, Sz: Integral points on hyperelliptic curves. Algebra Number Theory 2, 859–885 (2008)
Cohen, H.: Advanced Topics in Computational Algebraic Number Theory, vol. 193. Springer–Verlag, Berlin (2000)
David, A.: Caractère d’isogénie et crittéres d’irréductibilité, arXiv:1103.3892v2 [math.NT]
Dembélé, L., Voight, J.: Explicit methods for Hilbert modular forms, In Elliptic curves, Hilbert modular forms and Galois deformations, Adv. Courses Math. CRM Barcelona, pp. 135–198. Birkhäuser/Springer, Basel (2013)
Freitas, N., Le Hung, B., Siksek, S.: Elliptic curves over real quadratic fields are modular. Invent. Math. 201, 159–206 (2015)
Freitas, N., Siksek, S.: The asymptotic Fermat’s last theorem for five-sixths of real quadratic fields. Compos. Math. 151, 1395–1415 (2015)
Freitas, N., Siksek, S.: Criteria for the irreducibility of mod \(p\) representations of Frey curves. J. Théor. Nombres Bordeaux 27, 67–76 (2015)
Kraus, A., Oesterlé, J.: Sur une question de B. Mazur. Math. Ann 293, 259–275 (2002)
Kraus, A.: Sur l’équation \(a^{3}+b^{3} = c^{p}\). Exp. Math. 7, 1–13 (1998)
Laurent, M.: Linear forms in two logarithms and interpolation determinants. II. Acta Arith. 133, 325–348 (2008)
Matveev, E.: An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II. Izv. Math 64, 1217–1269 (2000)
Mignotte, M.: A kit on linear forms in three logarithms, p. 45. http://www-irma.u-strasbg.fr/~bugeaud/travaux/kit.ps
Momose, F.: Isogenies of prime degree over number fields. Compos. Math. 97, 329–348 (1995)
Pethő, A.: Perfect powers in second order linear recurrences. J. Number Theory 15, 5–13 (1982)
Shorey, T., Stewart, C.L.: On the Diophantine equation \(ax^{2t}+bx^ty+cy^2=d\) and pure powers in recurrence sequences. Math. Scand. 52, 24–36 (1983)
Silverman, J.H.: Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics 151. Springer-Verlag, New York (1994)
Smart, N.P.: The Algorithmic Resolution of Diophantine Equations. Cambridge University Press, Cambridge (1998)
Smart, N.P., Stephens, N.M.: Integral points on elliptic curves over number fields. Math. Proc. Camb. Philos. Soc. 122, 9–16 (1997)