Zeros of recurrence sequences

Abstract

We give an upper bound for the number of zeros of recurrence sequences defined over an algebraic number field in terms of their order, the degree of their field of definition and the number of prime ideal divisors of the characteristic roots of the sequence.

Decomposable form equations

“…[15]). The finiteness result quoted after Theorem K, and therefore the implication (i) ===> (ii) in Theorem K can be deduced from the following finiteness theorem on S-unit equations which was established independently by van der Poorten and Schlickewei [49] and Evertse [13]. In [15] (see also [16]) it has been pointed out that the implication (i) ===> (ii) of Theorem K is in fact equivalent to the above theorem on S-unit equations.…”

On the number of distinct prime factors of a sum of super-powers

“…(A recurrence is non-degenerate if its characteristic polynomial has at least two distinct non-zero complex roots and the ratio of any two distinct non-zero characteristic roots is not a root of unity.) More specifically, it was shown by van der Poorten and Schlickewei [14] and, independently, by Evertse [4, Corollary 3], using Schlickewei's p-adic analogue of Schmidt's Subspace Theorem [7], that the greatest prime factor of u n tends to ∞ as n → ∞. In a similar note, effective lower bounds on the greatest prime divisor and on the greatest square-free factor of a sequence of type (3) were obtained under mild assumptions by Shparlinski [10] and Stewart [11][12][13], based on variants of Baker's theorem on linear forms in the logarithms of algebraic numbers [2].…”

Equations 𝑎𝑢^{𝑙}_{𝑛}=𝑏𝑢^{𝑘}_{𝑚} satisfied by members of recurrence sequences

“…In the case when the vn are algebraic, Schlickewei [5] and van der Poorten and Schlickewei [4] have given a quantitative version of this result. Evertse [ 1 ] proved that the equation vn = vm admits only finitely many solutions n ^ m with n, m > 0.…”