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 pp--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 $p$--adic numbers $\mathbb Q_p$, also MCFs have been recently introduced in $\mathbb Q_p$ together to a $p$--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 $p$--adic Jacobi--Perron algorithm our results are obtained by exploiting properties of some auxiliary integer sequences. Moreover, it is known that a finite $p$--adic MCF represents $\mathbb Q$--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 $p$--adic valuations equal to $-1$. Finally, we show that a periodic MCF of dimension $m$ converges to algebraic irrationals of degree less or equal than $m+1$ and for the case $m=2$ 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

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 $$P^2 - D Q^2 = 1$$ where $D$ is a given integer polynomial and the solutions $P, Q$ must be integer polynomials. A classical paper of Nathanson \cite{Nat} solved it when $D(x) = x^2 + d$. 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 $D(x) = f^2(x) + d$, for any $f \in \mathbb Z[X]$ and $d \in \mathbb Z$, 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 $A$ is defined as $R(A) := \left\{a/b : a,b \in A,\; b \neq 0\right\}$. We consider the case in which $A$ is the image of $\mathbb{Z}^+$ under a polynomial $f \in \mathbb{Z}[X]$, and we give some conditions under which $R(A)$ is dense in $\mathbb{Q}_p$. Then, we apply these results to determine when $R(S_m^n)$ is dense in $\mathbb{Q}_p$, where $S_m^n$ is the set of numbers of the form $x_1^n + \cdots + x_m^n$, with $x_1, \dots, x_m \geq 0$ integers. This allows us to answer a question posed in [Garcia et al., $p$-adic quotient sets, Acta Arith. 179, 163-184]. We end leaving an open question