125 research outputs found
Linear recurrence sequences and periodicity of multidimensional continued fractions
Multidimensional continued fractions generalize classical continued fractions
with the aim of providing periodic representations of algebraic irrationalities
by means of integer sequences. However, there does not exist any algorithm that
provides a periodic multidimensional continued fraction when algebraic
irrationalities are given as inputs. In this paper, we provide a
characterization for periodicity of Jacobi--Perron algorithm by means of linear
recurrence sequences. In particular, we prove that partial quotients of a
multidimensional continued fraction are periodic if and only if numerators and
denominators of convergents are linear recurrence sequences, generalizing
similar results that hold for classical continued fractions
On the finiteness and periodicity of the --adic Jacobi--Perron algorithm
Multidimensional continued fractions (MCFs) were introduced by Jacobi and
Perron in order to obtain periodic representations for algebraic irrationals,
as it is for continued fractions and quadratic irrationals. Since continued
fractions have been also studied in the field of --adic numbers , also MCFs have been recently introduced in together to a
--adic Jacobi--Perron algorithm. In this paper, we address th study of two
main features of this algorithm, i.e., finiteness and periodicity. In
particular, regarding the finiteness of the --adic Jacobi--Perron algorithm
our results are obtained by exploiting properties of some auxiliary integer
sequences. Moreover, it is known that a finite --adic MCF represents
--linearly dependent numbers. We see that the viceversa is not
always true and we prove that in this case infinite partial quotients of the
MCF have --adic valuations equal to . Finally, we show that a periodic
MCF of dimension converges to algebraic irrationals of degree less or equal
than and for the case we are able to give some more detailed
results
An efficient and secure RSA--like cryptosystem exploiting R\'edei rational functions over conics
We define an isomorphism between the group of points of a conic and the set
of integers modulo a prime equipped with a non-standard product. This product
can be efficiently evaluated through the use of R\'edei rational functions. We
then exploit the isomorphism to construct a novel RSA-like scheme. We compare
our scheme with classic RSA and with RSA-like schemes based on the cubic or
conic equation. The decryption operation of the proposed scheme turns to be two
times faster than RSA, and involves the lowest number of modular inversions
with respect to other RSA-like schemes based on curves. Our solution offers the
same security as RSA in a one-to-one communication and more security in
broadcast applications.Comment: 18 pages, 1 figur
On the periodic writing of cubic irrationals and a generalization of Rédei functions
In this paper, we provide a periodic representation (by means of periodic rational or integer sequences) for any cubic irrationality. In particular, for a root α of a cubic polynomial with rational coefficients, we study the Cerruti polynomials [Formula: see text], and [Formula: see text], which are defined via [Formula: see text] Using these polynomials, we show how any cubic irrational can be written periodically as a ternary continued fraction. A periodic multidimensional continued fraction (with pre-period of length 2 and period of length 3) is proved convergent to a given cubic irrationality, by using the algebraic properties of cubic irrationalities and linear recurrent sequences.</jats:p
A note on the use of Rédei polynomials for solving the polynomial Pell equation and its generalization to higher degrees
The polynomial Pell equation is where is a given
integer polynomial and the solutions must be integer polynomials. A
classical paper of Nathanson \cite{Nat} solved it when . We
show that the R\'edei polynomials can be used in a very simple and direct way
for providing these solutions. Moreover, this approach allows to find all the
integer polynomial solutions when , for any and , generalizing the result of Nathanson. We are also
able to find solutions of some generalized polynomial Pell equations
introducing an extension of R\'edei polynomials to higher degrees
Squaring the magic squares of order 4
In this paper, we present the problem of counting magic squares and we focus
on the case of multiplicative magic squares of order 4. We give the exact
number of normal multiplicative magic squares of order 4 with an original and
complete proof, pointing out the role of the action of the symmetric group.
Moreover, we provide a new representation for magic squares of order 4. Such
representation allows the construction of magic squares in a very simple way,
using essentially only five particular 4X4 matrices
On the p-adic denseness of the quotient set of a polynomial image
The quotient set, or ratio set, of a set of integers is defined as . We consider the case in which
is the image of under a polynomial ,
and we give some conditions under which is dense in .
Then, we apply these results to determine when is dense in
, where is the set of numbers of the form , with integers. This allows us to answer a
question posed in [Garcia et al., -adic quotient sets, Acta Arith. 179,
163-184]. We end leaving an open question
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