scholarly journals Norm form equations and linear divisibility sequences

2021 ◽  
pp. 1-21
Author(s):  
Elisa Bellah
Keyword(s):  
Linear Recurrence ◽  
Diophantine Problem ◽  
Norm Form ◽  
Integer Solutions ◽  
Recurrence Sequences ◽  
Coefficient Ring

Finding integer solutions to norm form equations is a classical Diophantine problem. Using the units of the associated coefficient ring, we can produce sequences of solutions to these equations. It is known that these solutions can be written as tuples of linear recurrence sequences. We show that for certain families of norm forms defined over quartic fields, there exist integrally equivalent forms making any one fixed coordinate sequence a linear divisibility sequence.

The Quotient Problem for Entire Functions

2018 ◽  
Vol 62 (3) ◽  
pp. 479-489 ◽  
Author(s):  
Ji Guo
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Linear Recurrence ◽  
Natural Numbers ◽  
Complex Situation ◽  
Index Set ◽  
Diophantine Problem ◽  
Divisibility Problem ◽  
Recurrence Sequences

AbstractLet $\{\mathbf{F}(n)\}_{n\in \mathbb{N}}$ and $\{\mathbf{G}(n)\}_{n\in \mathbb{N}}$ be linear recurrence sequences. It is a well-known Diophantine problem to determine the finiteness of the set ${\mathcal{N}}$ of natural numbers such that their ratio $\mathbf{F}(n)/\mathbf{G}(n)$ is an integer. In this paper we study an analogue of such a divisibility problem in the complex situation. Namely, we are concerned with the divisibility problem (in the sense of complex entire functions) for two sequences $F(n)=a_{0}+a_{1}f_{1}^{n}+\cdots +a_{l}f_{l}^{n}$ and $G(n)=b_{0}+b_{1}g_{1}^{n}+\cdots +b_{m}g_{m}^{n}$, where the $f_{i}$ and $g_{j}$ are nonconstant entire functions and the $a_{i}$ and $b_{j}$ are non-zero constants except that $a_{0}$ can be zero. We will show that the set ${\mathcal{N}}$ of natural numbers such that $F(n)/G(n)$ is an entire function is finite under the assumption that $f_{1}^{i_{1}}\cdots f_{l}^{i_{l}}g_{1}^{j_{1}}\cdots g_{m}^{j_{m}}$ is not constant for any non-trivial index set $(i_{1},\ldots ,i_{l},j_{1},\ldots ,j_{m})\in \mathbb{Z}^{l+m}$.

Divisor Sums of Generalised Exponential Polynomials

1996 ◽  
Vol 39 (1) ◽  
pp. 35-46 ◽  
Author(s):  
G. R. Everest ◽  
I. E. Shparlinski
Keyword(s):  
Prime Ideal ◽  
Algebraic Integer ◽  
Linear Recurrence ◽  
Exponential Polynomial ◽  
Exponential Polynomials ◽  
Recurrence Sequence ◽  
Special Cases ◽  
Divisor Sums ◽  
Recurrence Sequences

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.

scholarly journals Decomposable polynomials in second order linear recurrence sequences

2018 ◽  
Vol 159 (3-4) ◽  
pp. 321-346 ◽  
Author(s):  
Clemens Fuchs ◽  
Christina Karolus ◽  
Dijana Kreso
Keyword(s):  
Second Order ◽  
Linear Recurrence ◽  
Order Linear ◽  
Recurrence Sequences

scholarly journals Linear combinations of prime powers in binary recurrence sequences

2017 ◽  
Vol 13 (02) ◽  
pp. 261-271 ◽  
Author(s):  
Csanád Bertók ◽  
Lajos Hajdu ◽  
István Pink ◽  
Zsolt Rábai
Keyword(s):  
Linear Combination ◽  
The Other ◽  
Linear Recurrence ◽  
Linear Combinations ◽  
Other Hand ◽  
Sums Of Powers ◽  
Recurrence Sequences

We give finiteness results concerning terms of linear recurrence sequences having a representation as a linear combination, with fixed coefficients, of powers of fixed primes. On one hand, under certain conditions, we give effective bounds for the terms of binary recurrence sequences with such a representation. On the other hand, in the case of some special binary recurrence sequences, all terms having a representation as sums of powers of [Formula: see text] and [Formula: see text] are explicitly determined.

scholarly journals ‐PARTS OF TERMS OF INTEGER LINEAR RECURRENCE SEQUENCES

Mathematika ◽  
2017 ◽  
Vol 63 (3) ◽  
pp. 840-851
Author(s):  
Yann Bugeaud ◽  
Jan‐Hendrik Evertse
Keyword(s):  
Linear Recurrence ◽  
Recurrence Sequences

scholarly journals General Linear Recurrence Sequences and Their Convolution Formulas

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 132 ◽  
Author(s):  
Paolo Emilio Ricci ◽  
Pierpaolo Natalini
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Second Order ◽  
General Linear ◽  
Linear Recurrence ◽  
Order Linear ◽  
Matrix Polynomials ◽  
Recurrence Sequences ◽  
Straightforward Extension

We extend a technique recently introduced by Chen Zhuoyu and Qi Lan in order to find convolution formulas for second order linear recurrence polynomials generated by 1 1 + a t + b t 2 x . The case of generating functions containing parameters, even in the numerator is considered. Convolution formulas and general recurrence relations are derived. Many illustrative examples and a straightforward extension to the case of matrix polynomials are shown.

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