Binary Recurrence Sequences Research Articles

Divisor Sums of Generalised Exponential Polynomials

AbstractA study is made of sums of reciprocal norms of integral and prime ideal divisors of algebraic integer values of a generalised exponential polynomial. This includes the important special cases of linear recurrence sequences and general sums of S-units. In the case of an integral binary recurrence sequence, similar (but stronger) results were obtained by P. Erdős, P. Kiss and C. Pomerance.

On binary recurrence sequences

On binary recurrence sequences

Linear forms in Lucas numbers

If (u m) m∈ N 0 denotes a Lucas sequence, i.e. a binary integer recurrence sequence with initial values u 0=0 and u 1=1, then the equation ku m = lu n with k,l∈ Z⧹{0} and max{ m, n}≥5 can be valid only for finitely many Lucas sequences with coprime roots and finitely many indices m,n∈ N , which can - both - be effectively bounded. This yields lower bounds for | ku m − lu n |. In the same way the equation u n = l can be considered, and this gives a partial answer to a conjecture of Beukers concerning multiplicities of binary recurrences. The proofs depend on estimates for linear forms in logarithms and on bounds for the solutions of equations in binary forms.

Products of Prime Powers in Binary Recurrence Sequences Part I: The Hyperbolic Case, with an Application to the Generalized Ramanujan-Nagell Equation

We show how the Gelfond-Baker theory and diophantine approximation techniques can be applied to solve explicitly the diophantine equation [formula] (where [formula] is a binary recurrence sequence with positive discriminant), for arbitrary values of the parameters. We apply this to the equation [formula] , which is a generalization of the Ramanujan-Nagell equation [formula]. We present algorithms to reduce upper bounds for the solutions of these equations. The algorithms are easy to translate into computer programs. We present an example which shows that in practice the method works well.

Products of Prime Powers in Binary Recurrence Sequences Part II: The Elliptic Case, with an Application to a Mixed Quadratic-Exponential Equation

In Part I the diophantine equation [formula] was studied, where [formula] is a linear binary recurrence sequence with positive discriminant. In this second part we extend this to negative discriminants. We use the p-adic and complex Gelfond-Baker theory to find explicit upper bounds for the solutions of the equation. We give algorithms to reduce those bounds, based on diophantine approximation techniques. Thus we have a method to solve the equation completely for arbitrary values of the parameters. We give an application to a quadratic-exponential equation.

Products of prime powers in binary recurrence sequences. I. The hyperbolic case, with an application to the generalized Ramanujan-Nagell equation

We show how the Gelfond-Baker theory and diophantine approximation techniques can be applied to solve explicitly the diophantine equation G n = w p 1 m 1 ⋯ p t m t {G_n} = wp_1^{{m_1}} \cdots p_t^{{m_t}} (where { G n } n = 0 ∞ \{ {G_n}\} _{n = 0}^\infty is a binary recurrence sequence with positive discriminant), for arbitrary values of the parameters. We apply this to the equation x 2 + k = p 1 z 1 ⋯ p t z t {x^2} + k = p_1^{{z_1}} \cdots p_t^{{z_t}} , which is a generalization of the Ramanujan-Nagell equation x 2 + 7 = 2 z {x^2} + 7 = {2^z} . We present algorithms to reduce upper bounds for the solutions of these equations. The algorithms are easy to translate into computer programs. We present an example which shows that in practice the method works well.

Products of prime powers in binary recurrence sequences. II. The elliptic case, with an application to a mixed quadratic-exponential equation

In Part I the diophantine equation G n = w p 1 m 1 ⋯ p t m t {G_n} = wp_1^{{m_1}} \cdots p_t^{{m_t}} was studied, where { G n } n = 0 ∞ \{ {G_n}\} _{n = 0}^\infty is a linear binary recurrence sequence with positive discriminant. In this second part we extend this to negative discriminants. We use the p-adic and complex Gelfond-Baker theory to find explicit upper bounds for the solutions of the equation. We give algorithms to reduce those bounds, based on diophantine approximation techniques. Thus we have a method to solve the equation completely for arbitrary values of the parameters. We give an application to a quadratic-exponential equation.