On numbers nn dividing the nnth term of a linear recurrence

Here, we give upper and lower bounds on the count of positive integers $n\le x$ dividing the $n$th term of a nondegenerate linearly recurrent sequence with simple roots

Primality tests, linear recurrent sequences and the Pell equation

We study new primality tests based on linear recurrent sequences of degree two exploiting a matricial approach. The classical Lucas test arises as a particular case and we see how it can be easily improved. Moreover, this approach shows clearly how the Lucas pseudoprimes are connected to the Pell equation and the Brahamagupta product. We also introduce a new specific primality test, which we will call generalized Pell test. We perform some numerical computations on the new primality tests and, for the generalized Pell test, we do not any pseudoprime up to $10^{10}$

SOME CONNECTIONS BETWEEN THE SMARANDACHE FUNCTION AND THE FIBONACCI SEQUENCE

This paper is aimed to provide generalizations of the Smarandache function. They will be constructed by means of sequences more general than the sequence of the factorials. Such sequences are monotonously convergent to zero sequences and divisibility sequences (in particular the Fibonacci sequence)

Primality tests, linear recurrent sequences and the Pell equation

4We study new primality tests based on linear recurrent sequences of degree two exploiting a matrix approach. The classical Lucas test arises as a particular case and we see how it can be easily improved. Moreover, this approach shows clearly how the Lucas pseudoprimes are connected to the Pell equation and the Brahamagupta product. We also introduce two new specific primality tests, which we will call generalized Lucas test and generalized Pell test. We perform some numerical computations on the new primality tests and we do not find any pseudoprime up to 238. Moreover, we combined the generalized Lucas test with the Fermat test up to 264 and we did not find any composite number that passes the test. We get the same result using the generalized Pell test.partially_openembargoed_20220207Bazzanella, Danilo; Di Scala, Antonio; Dutto, Simone; Murru, NadirBazzanella, Danilo; Di Scala, Antonio; Dutto, Simone; Murru, Nadi

On Generalized Lucas Pseudoprimality of Level k

We investigate the Fibonacci pseudoprimes of level k, and we disprove a statement concerning the relationship between the sets of different levels, and also discuss a counterpart of this result for the Lucas pseudoprimes of level k. We then use some recent arithmetic properties of the generalized Lucas, and generalized Pell–Lucas sequences, to define some new types of pseudoprimes of levels k+ and k− and parameter a. For these novel pseudoprime sequences we investigate some basic properties and calculate numerous associated integer sequences which we have added to the Online Encyclopedia of Integer Sequences.N/

On some new arithmetic properties of the generalized Lucas sequences

Some arithmetic properties of the generalized Lucas sequences are studied, extending a number of recent results obtained for Fibonacci, Lucas, Pell, and Pell–Lucas sequences. These properties are then applied to investigate certain notions of Fibonacci, Lucas, Pell, and Pell–Lucas pseudoprimality, for which we formulate some conjectures.N/

Recurrent Sequences and Cryptography

Fibonacci numbers are defined as recursively as F_(n+1)=F_n+F_(n-1) with initial conditions F_1=F_2=1. Lucas numbers also enjoys the same recurrence relation but with different initial conditions L_1=1,L_2=3. Large prims are very useful in public key cryptography. Lucas numbers can also be exploited for these purpose