77 research outputs found

    An Active Learning Activity for the Construction of a Finite-Field Slide Rule for Undergraduate Students

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    Math manipulatives and physical math activities offer numerous benefits in the learning process in all educational levels: they provide concrete representations of abstract concepts, thus helping more students to understand them, and they are an opportunity for students to explore and test their understanding of mathematical concepts. Moreover, in active learning activities conducted with tangible objects, students are physically engaged in the lessons, and are given a fun way to practice their math skills, which contributes with retention and positive feeling. In this article, an active teaching activity aimed at first-year college students is presented, designed to deepen the understanding of finite fields through the construction of a slide rule. The tool presented is easy to make and can be used effectively in a short time. The activity described was carried out as part of a larger workshop on modular arithmetic and the basics of cryptography offered to first-year engineering students at Politecnico di Torino

    Periodic representations and rational approximations of square roots

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    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

    Polynomial sequences on quadratic curves

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    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

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    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 3×33 \times 3 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

    Linear divisibility sequences and Salem numbers

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    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

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    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

    The Biharmonic mean

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    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
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