Diophantine equations with linear recurrences An overview of some recent progress

Zannier, Umberto

Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 423-435.

Nous discutons quelques problèmes habituels concernant l’arithmétique des suites récurrentes linéaires. Après avoir brièvement rappelé les questions et résultats anciens concernant les zéros, nous nous focalisons sur les progrès récents pour le “problème quotient” (resp. “problème de la racine $d$-ième”), qui, pour faire court, demande si l’intégralité des valeurs du quotient (resp. racine $d$-ième) de deux (resp. d’une) suites récurrentes linéaires entraine que ce quotient (resp. racine $d$-ième) est lui-même une suite récurrente linéaire. Nous relions également ces questions à certaines équations diophantiennes naturelles, qui par ailleurs proviennent du cas non résolu le plus simple de la conjecture de Vojta sur les points entiers des variétés algébriques.

We shall discuss some known problems concerning the arithmetic of linear recurrent sequences. After recalling briefly some longstanding questions and solutions concerning zeros, we shall focus on recent progress on the so-called “quotient problem” (resp. "$d$-th root problem"), which in short asks whether the integrality of the values of the quotient (resp. $d$-th root) of two (resp. one) linear recurrences implies that this quotient (resp. $d$-th root) is itself a recurrence. We shall also relate such questions with certain natural diophantine equations, which in turn come from the simplest unknown cases of Vojta’s conjecture for integral points on algebraic varieties.

[BCZ] Y. Bugeaud, P. Corvaja, U. Zannier, An upper bound for the G.C.D. of $a^n-1$ and $b^n-1$. Math. Z. 243 (2003), 79–84. | MR | Zbl

[CZ1] P. Corvaja, U. Zannier, Diophantine equations with power sums and Universal Hilbert Sets. Indag. Mathem., N.S. 9 (3) (1998), 317–332. | MR | Zbl

[CZ2] P. Corvaja, U. Zannier, Finiteness of integral values for the ratio of two linear recurrences. Invent. Math. 149 (2002), 431–451. | MR | Zbl

[CZ3] P. Corvaja, U. Zannier, On the greatest prime factor of $(ab+1)(ac+1)$. Proc. Amer. Math. Soc. 131 (2003), 1705–1709. | MR | Zbl

[CZ4] P. Corvaja, U. Zannier, On the length of the continued fraction for values of quotients of power sums. Preprint 2003, to appear on Journal de Théorie des nombres de Bordeaux. | Numdam | MR | Zbl

[CZ5] P. Corvaja, U. Zannier, Some New Applications of the Subspace Theorem, Compositio Math. 131 (2002), 319–340. | MR | Zbl

[CZ6] P. Corvaja, U. Zannier, On the diophantine equation $f(a^m,y)=b^n$. Acta Arith. 94.1 (2000), 25–40. | MR | Zbl

[DZ] P. Dèbes, U. Zannier, Universal Hilbert Subsets. Math. Proc. Camb. Phil. Soc. 124 (1998), 127–134. | MR | Zbl

[HS] M. Hindry, J.H. Silverman, Diophantine Geometry. Springer-Verlag, 2000. | MR | Zbl

[PeZ] A. Perelli, U. Zannier, Arithmetic properties of certain recurrent sequences. J. Austral. Math. Soc. (A) 37 (1984), 4–16. | MR | Zbl

[vdP] A.J. van der Poorten, Some facts that should be better known, especially about rational functions. In Number Theory and Applications, (Banff, AB 1988), 497–528, Kluwer Acad. Publ., Dordrecht, 1989. | MR | Zbl

[vdP2] A.J. van der Poorten, Solution de la conjecture de Pisot sur le quotient de Hadamard de deux fractions rationnelles. C. R. Acad. Sci. Paris 306 (1988), 97–102. | MR | Zbl

[Po] Y. Pourchet, Solution du problème arithmétique du quotient de Hadamard de deux fractions rationnelles. C. R. Acad. Sci. Paris 288, Série $A$ (1979), 1055–1057. | MR | Zbl

[R] R. Rumely, Note on van der Poorten’s proof of the Hadamard quotient theorem I, II. In: Séminaire de Théorie des nombres de Paris 1986-87, 349–409, Progress in Math. 75, Birkhäuser, Boston, 1988. | MR | Zbl

[S] W.M. Schmidt, The zero multiplicity of linear recurrence sequences. Acta Math. 182 (1999), 243–282. | MR | Zbl

[S2] W.M. Schmidt, Linear Recurrence Sequences and Polynomial-Exponential Equations. In Diophantine Approximation, F. Amoroso, U. Zannier Eds., Proc. of the C.I.M.E. Conference, Cetraro (Italy), 2000, Springer-Verlag LNM 1819, 2003. | MR | Zbl

[ShSt] T.N. Shorey, C.L. Stewart, Pure Powers in Recurrence Sequences and Some Related Diophantine Equations. Journal of Number Theory 27 (1987), 324–352. | MR | Zbl

[ShT] T.N. Shorey, R. Tijdeman, Exponential Diophantine Equations. Camb. Univ. Press, 1986. | MR | Zbl

[Z] U. Zannier, Some applications of Diophantine Approximation to Diophantine Equations. Editrice Forum, Udine, Dicembre 2003.

[Z2] U. Zannier, A proof of Pisot $d^{th}$ root conjecture. Annals of Math. 151 (2000), 375–383. | EuDML | MR | Zbl