76 research outputs found
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
Linear divisibility sequences and Salem numbers
We study linear divisibility sequences of order 4, providing a
characterization by means of their characteristic polynomials and finding their
factorization as a product of linear divisibility sequences of order 2.
Moreover, we show a new interesting connection between linear divisibility
sequences and Salem numbers. Specifically, we generate linear divisibility
sequences of order 4 by means of Salem numbers modulo 1
Identities Involving Zeros of Ramanujan and Shanks Cubic Polynomials
In this paper we highlight the connection between Ramanujan cubic polynomials
(RCPs) and a class of polynomials, the Shanks cubic polynomials (SCPs), which
generate cyclic cubic fields. In this way we provide a new characterization for
RCPs and we express the zeros of any RCP in explicit form, using trigonometric
functions. Moreover, we observe that a cyclic transform of period three
permutes these zeros. As a consequence of these results we provide many new and
beautiful identities. Finally we connect RCPs to Gaussian periods, finding a
new identity, and we study some integer sequences related to SCPs
Polynomial sequences on quadratic curves
In this paper we generalize the study of Matiyasevich on integer points over
conics, introducing the more general concept of radical points. With this
generalization we are able to solve in positive integers some Diophantine
equations, relating these solutions by means of particular linear recurrence
sequences. We point out interesting relationships between these sequences and
known sequences in OEIS. We finally show connections between these sequences
and Chebyshev and Morgan-Voyce polynomials, finding new identities
Groups and monoids of Pythagorean triples connected to conics
We define operations that give the set of all Pythagorean triples a structure
of commutative monoid. In particular, we define these operations by using
injections between integer triples and matrices. Firstly, we
completely characterize these injections that yield commutative monoids of
integer triples. Secondly, we determine commutative monoids of Pythagorean
triples characterizing some Pythagorean triple preserving matrices. Moreover,
this study offers unexpectedly an original connection with groups over conics.
Using this connection, we determine groups composed by Pythagorean triples with
the studied operations
The Biharmonic mean
We briefly describe some well-known means and their properties, focusing on
the relationship with integer sequences. In particular, the harmonic numbers,
deriving from the harmonic mean, motivate the definition of a new kind of mean
that we call the biharmonic mean. The biharmonic mean allows to introduce the
biharmonic numbers, providing a new characterization for primes. Moreover, we
highlight some interesting divisibility properties and we characterize the
semi--prime biharmonic numbers showing their relationship with linear recurrent
sequences that solve certain Diophantine equations
Periodic representations and rational approximations of square roots
In this paper the properties of R\'edei rational functions are used to derive
rational approximations for square roots and both Newton and Pad\'e
approximations are given as particular cases. As a consequence, such
approximations can be derived directly by power matrices. Moreover, R\'edei
rational functions are introduced as convergents of particular periodic
continued fractions and are applied for approximating square roots in the field
of p-adic numbers and to study periodic representations. Using the results over
the real numbers, we show how to construct periodic continued fractions and
approximations of square roots which are simultaneously valid in the real and
in the p-adic field
- …