Some Monoapparitic Fourth Order Linear Divisibility Sequences
-
Hugh C. Williams
and
Richard K. Guy
Abstract.
A sequence of rational integers 
is said to be a divisibility
sequence if 
whenever 
. If the divisibility
sequence also satisfies a linear recurrence relation of
order k, it is said to be a linear divisibility sequence. The best
known example of a linear divisibility sequence of order 2 is the
Lucas sequence 
, one particular instance of which is the famous
Fibonacci sequence. In their extension of the Lucas functions to
order 4 linear recursions, Williams and Guy showed that the order
4 analog 
of can have no more than two ranks of
apparition for a given prime p and frequently has two such ranks,
unlike the situation for , which can only have one rank of
apparition. In this paper we investigate the problem of finding
those sequences which have only one rank of apparition for
any prime p.
© 2012 by Walter de Gruyter Berlin Boston
Articles in the same Issue
- Masthead
- Preface to the John Selfridge Memorial Issue
- Numbers with Integer Complexity Close to the Lower Bound
- On a Conjecture Regarding Balancing with Powers of Fibonacci Numbers
- Perfect Powers with Few Ternary Digits
- On Mullin's Second Sequence of Primes
- Log-Sine Evaluations of Mahler Measures, II
- On Odd Perfect Numbers and Even 3-Perfect Numbers
- Euler Pseudoprimes for Half of the Bases
- Odd Incongruent Restricted Disjoint Covering Systems
- Cubes in {0,1,...,n}3
- Sierpiński Numbers in Imaginary Quadratic Fields
- On a Partition Problem of Canfield and Wilf
- The 392 Problem
- Artin's Primitive Root Conjecture – A Survey
- Prime-Perfect Numbers
- Explicit Solutions of Certain Systems of Pell Equations
- The Search for Aurifeuillian-Like Factorizations
- Some Monoapparitic Fourth Order Linear Divisibility Sequences
- A Note from the Editors
Articles in the same Issue
- Masthead
- Preface to the John Selfridge Memorial Issue
- Numbers with Integer Complexity Close to the Lower Bound
- On a Conjecture Regarding Balancing with Powers of Fibonacci Numbers
- Perfect Powers with Few Ternary Digits
- On Mullin's Second Sequence of Primes
- Log-Sine Evaluations of Mahler Measures, II
- On Odd Perfect Numbers and Even 3-Perfect Numbers
- Euler Pseudoprimes for Half of the Bases
- Odd Incongruent Restricted Disjoint Covering Systems
- Cubes in {0,1,...,n}3
- Sierpiński Numbers in Imaginary Quadratic Fields
- On a Partition Problem of Canfield and Wilf
- The 392 Problem
- Artin's Primitive Root Conjecture – A Survey
- Prime-Perfect Numbers
- Explicit Solutions of Certain Systems of Pell Equations
- The Search for Aurifeuillian-Like Factorizations
- Some Monoapparitic Fourth Order Linear Divisibility Sequences
- A Note from the Editors
