Remarks on a conjecture on certain integer sequences (original) (raw)

Integer sequences and periodic points

Arithmetic properties of integer sequences counting periodic points are studied, and applied to the case of linear recurrence sequences, Bernoulli numerators, and Bernoulli denominators.

Periodicity of Balancing Numbers

Acta Mathematica Hungarica

The balancing numbers originally introduced by Behera and Panda as solutions of a Diophantine equation on triangular numbers possess many interesting properties. Many of these properties are comparable to certain properties of Fibonacci numbers, while some others are more interesting. Wall studied the periodicity of Fibonacci numbers modulo arbitrary natural numbers. The periodicity of balancing numbers modulo primes and modulo terms of certain sequences exhibits beautiful results, again, some of them are identical with corresponding results of Fibonacci numbers while some others are more interesting. An important observation concerning the periodicity of balancing numbers is that, the period of this sequence coincides with the modulus of congruence if the modulus is any power of 2. There are three known primes for which the periods of the sequence of balancing numbers modulo each prime are equal to the period modulo square of that prime, while for the Fibonacci sequence, till date ...

Sequences of integers, conjectures and new arithmetical tools

In three of my previous published books, namely “Two hundred conjectures and one hundred and fifty open problems on Fermat pseudoprimes”, “Conjectures on primes and Fermat pseudoprimes, many based on Smarandache function” and “Two hundred and thirteen conjectures on primes”, I showed my passion for conjectures on sequences of integers. In spite the fact that some mathematicians stubbornly understand mathematics as being just the science of solving and proving, my books of conjectures have been well received by many enthusiasts of elementary number theory, which gave me confidence to continue in this direction. Part One of this book brings together papers regarding conjectures on primes, twin primes, squares of primes, semiprimes, different types of pairs or triplets of primes, recurrent sequences, sequences of integers created through concatenation and other sequences of integers related to primes. Part Two of this book brings together several articles which present the notions of c-primes, m-primes, c-composites and m-composites (c/m-integers), also the notions of g-primes, s-primes, g-composites and s-composites (g/s-integers) and show some of the applications of these notions (because this is not a book structured unitary from the beginning but a book of collected papers, I defined the notions mentioned in various papers, but the best definition of them can be found in Addenda to the paper numbered tweny-nine), in the study of the squares of primes, Fermat pseudoprimes and generally in Diophantine analysis. Part Three of this book presents the notions of “Coman constants” and “Smarandache-Coman constants”, useful to highlight the periodicity of some infinite sequences of positive integers (sequences of squares, cubes, triangular numbers, polygonal numbers), respectively in the analysis of Smarandache concatenated sequences. Part Four of this book presents the notion of Smarandache-Coman sequences, id est sequences of primes formed through different arithmetical operations on the terms of Smarandache concatenated sequences. Part Five of this book presents the notion of Smarandache-Coman function, a function based on the well known Smarandache function which seems to be particularly interesting: beside other characteristics, it seems to have as values all the prime numbers and, more than that, they seem to appear, leaving aside the non-prime values, in natural order.

Sequences close to periodic (Muchnik A. A., Pritykin Y. L., Semenov A. L.)

Russian Mathematical Surveys, 2009

This paper is a survey of concepts and results connected with generalizations of the notion of a periodic sequence, both classical and new. The topics discussed relate to almost periodicity in such areas as combinatorics on words, symbolic dynamics, expressibility in logical theories, computability, Kolmogorov complexity, and number theory. Bibliography: 124 titles.

On certain positive integer sequences

2004

A survey of recent results in elementary number theory is presented in this paper. Special attention is given to structure and asymptotic properties of certain families of positive integers.

On Periodic 1-GENERALIZED Fibonacci Sequences

2003

1. INTRODUCTIONThe notion of an 1-generalized Fibonacci sequence has been introduced in [6], and studiedin [1], [7], [9]. This class of sequences de ned by linear recurrences of in nite order is anextension of the class of ordinary (weighted) r-generalized Fibonacci sequences (r-GFS, forshort) with r nite de ned by linear recurrences of r

The encyclopedia of integer sequences

1996

This paper gives a brief description of the author's database of integer sequences, now over 35 years old, together with a selection of a few of the most interesting sequences in the table. Many unsolved problems are mentioned. This paper was published (in a somewhat different form) in Sequences and their Applications

Periodicity of Generalized Fibonacci-like Sequences

Generalized Fibonacci-like sequences appear in finite difference approximations of the Partial Differential Equations based upon replacing partial differential equations by finite difference equations. This paper studies properties of the generalized Fibonacci-like sequence í µí°¹í µí°¹ í µí±›í µí±›+2 = í µí°´í µí°´+ í µí°µí µí°µí µí°¹í µí°¹ í µí±›í µí±›+1 + í µí° ¶í µí° ¶í µí°¹í µí°¹ í µí±›í µí±›. It is shown that this sequence is periodic with period T>2, if C=-1, |B|<2.