Dynamical system proof of Fermat’s little theorem: An alternative approach

Fermat’s little theorem has been proved using different mathematical approaches, which majority of them are based on number theory. These approaches have only exposed the usability of Fermat’s little theorem for logical, linear and structural predictions. Only small numbers of attempts had only been made to proof Fermat’s little theorem from other perspectives. This paper exhibits an alternative approach to proof the Fermat’s little theorem via dynamical system. Two lemmas are proven with respect to a redefined function, Tn (x) in order to achieve the task.

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KALKULUS DIFERENSIAL DAN INTEGRAL OLEH FERMAT

JURNAL PIJAR MIPA ◽

10.29303/jpm.v7i1.91 ◽

2012 ◽

Vol 7 (1) ◽

Author(s):

Laila Hayati ◽

Mamika Ujianita Romdhini

Keyword(s):

Number Theory ◽

Historical Development ◽

Tangent Line ◽

Ancient Times ◽

Fermat’S Little Theorem

Abstrak. Dalam kuliah kalkulus modern, materi tentang pendifferensialan (turunan fungsi) dan konstruksi garis singgung terhadap suatu kurva diberikan terlebih dahulu daripada materi integral dan penentuan luas daerah di bawah suatu kurva. Hal ini berlawanan dengan urutan sejarah perkembangannnya. Penentuan luas daerah yang dibatasi oleh beberapa kurva telah ditemukan pada zaman kuno. Dalam tulisan ini membahas awal konstruksi garis singgung dan penentuan luas daerah yang dibatasi oleh suatu kurva yang pertama kali dibahas oleh Fermat. Kerja Fermat telah memberikan dasar bagi konsep kalkulus modern, khususnya pendifferensialan dan integral. Selain itu, Fermat dikenal sebagai orang yang memiliki kemampuan luar biasa dalam teori bilangan, antara lain dengan Fermat’s Little Theorem dan Fermat’s Last Theorem.Kata kunci: konstruksi garis singgung, luas daerah, differensial, dan integral, teori fermat. Abstrak. In modern calculus course, the material on derivative of the function and the construction of the tangent to the curve given first than the material on the integral and determining the area under a curve. This is contrary to the historical development. Determination of the area has been limited by several curves have been found in ancient times. In this paper discusses the start of construction of the tangent line and determining the area bounded by a curve that was first discussed by Fermat. Work Fermat has provided the basis for the concept of modern calculus, especially derivative and integral. In addition, Fermat is known as a person who has a remarkable ability in number theory, among others, by Fermat's Little Theorem and Fermat's Last Theorem.Keywords: construction of a tangent, wide areas, derivative and integral, Fermat Theory.

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Dilating Covariant Representations of a Semigroup Dynamical System Arising from Number Theory

Integral Equations and Operator Theory ◽

10.1007/s00020-015-2263-0 ◽

2015 ◽

Vol 84 (2) ◽

pp. 217-233

Author(s):

Jaspar Wiart

Keyword(s):

Dynamical System ◽

Number Theory ◽

Covariant Representations

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Fermat’s Little Theorem via Divisibility of Newton’s Binomial

Formalized Mathematics ◽

10.1515/forma-2015-0018 ◽

2015 ◽

Vol 23 (3) ◽

pp. 215-229 ◽

Cited By ~ 1

Author(s):

Rafał Ziobro

Keyword(s):

Number Theory ◽

Solving Equations ◽

Fermat’S Little Theorem

Abstract Solving equations in integers is an important part of the number theory [29]. In many cases it can be conducted by the factorization of equation’s elements, such as the Newton’s binomial. The article introduces several simple formulas, which may facilitate this process. Some of them are taken from relevant books [28], [14]. In the second section of the article, Fermat’s Little Theorem is proved in a classical way, on the basis of divisibility of Newton’s binomial. Although slightly redundant in its content (another proof of the theorem has earlier been included in [12]), the article provides a good example, how the application of registrations could shorten the length of Mizar proofs [9], [17].

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Ball-avoiding theorems

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700001000 ◽

2000 ◽

Vol 20 (6) ◽

pp. 1821-1849 ◽

Cited By ~ 2

Author(s):

DOMOKOS SZÁSZ

Keyword(s):

Dynamical System ◽

Number Theory ◽

Phase Space ◽

Hausdorff Dimension ◽

Open Subset ◽

Small Subset ◽

Interested Reader ◽

Rigorous Results ◽

Hard Ball

Consider a nice hyperbolic dynamical system (singularities not excluded). Statements about the topological smallness of the subset of orbits, which avoid an open subset of the phase space (for every moment of time, or just for a not too small subset of times), play a key role in showing hyperbolicity or ergodicity of semi-dispersive billiards, especially, of hard-ball systems. As well as surveying the characteristic results, called ball-avoiding theorems, and giving an idea of the methods of their proofs, their applications are also illustrated. Furthermore, we also discuss analogous questions (which had arisen, for instance, in number theory), when the Hausdorff dimension is taken instead of the topological one. The answers strongly depend on the notion of dimension which is used. Finally, ball-avoiding subsets are naturally related to repellers extensively studied by physicists. For the interested reader we also sketch some analytical and rigorous results about repellers and escape times.

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The Plaid Model

10.23943/princeton/9780691181387.001.0001 ◽

2019 ◽

Author(s):

Richard Evan Schwartz

Keyword(s):

Dynamical System ◽

Number Theory ◽

Continued Fractions ◽

Planetary Motion ◽

Elementary Number Theory ◽

Convex Shape ◽

Tile System ◽

Outer Billiards ◽

Combinatorial Model ◽

Self Similar

Outer billiards provides a toy model for planetary motion and exhibits intricate and mysterious behavior even for seemingly simple examples. It is a dynamical system in which a particle in the plane moves around the outside of a convex shape according to a scheme that is reminiscent of ordinary billiards. This book provides a combinatorial model for orbits of outer billiards on kites. The book relates these orbits to such topics as polytope exchange transformations, renormalization, continued fractions, corner percolation, and the Truchet tile system. The combinatorial model, called “the plaid model,” has a self-similar structure that blends geometry and elementary number theory. The results were discovered through computer experimentation and it seems that the conclusions would be extremely difficult to reach through traditional mathematics. The book includes an extensive computer program that allows readers to explore the materials interactively and each theorem is accompanied by a computer demonstration.

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