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Academic literature on the topic 'Fermat's equation'

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Published: 4 June 2021

Last updated: 2 February 2022

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Journal articles on the topic "Fermat's equation"

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Dolan, Stan. "Pell's equation and Fermat." Mathematical Gazette 96, no. 535 (March 2012): 66–70. http://dx.doi.org/10.1017/s0025557200003971.

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Given that it took until the 19th century for the case n = 5 of Fermat's last theorem to be settled, it is not surprising that Fermat's claim of having a proof for all exponents greater than 2 is nowadays treated with considerable scepticism. However, perhaps the most important aspect of his claim has been the impetus it has given to the development of mathematical techniques over the three and a half centuries leading up to the proof by Wiles and Taylor [1, 2]. Not least amongst these techniques has been Fermat's own idea of proof by descente infinie [3,4].

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Dem'yanenko, V. A. "ELLIPTIC FUNCTIONS AND FERMAT'S EQUATION." Russian Academy of Sciences. Sbornik Mathematics 77, no. 1 (February 28, 1994): 11–23. http://dx.doi.org/10.1070/sm1994v077n01abeh003426.

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Zhong, Cui-Xiang. "On Fermat's equation with prime power exponents." Acta Arithmetica 59, no. 1 (1991): 83–86. http://dx.doi.org/10.4064/aa-59-1-83-86.

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Cohn, J. H. E. "The Diophantine equation x2+3 = yn." Glasgow Mathematical Journal 35, no. 2 (May 1993): 203–6. http://dx.doi.org/10.1017/s0017089500009757.

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Many special cases of the equation x2+C= yn where x and y are positive integers and n≥3 have been considered over the years, but most results for general n are of fairly recent origin. The earliest reference seems to be an assertion by Fermat that he had shown that when C=2, n=3, the only solutions are given by x = 5, y = 3; a proof was published by Euler [1]. The first result for general n is due to Lebesgue [2] who proved that when C = 1 there are no solutions. Nagell [4] generalised Fermat's result and proved that for C = 2 the equation has no solution other than x = 5, y = 3, n = 3. He also showed [5] that for C = 4 the equation has no solution except x = 2, y = 2, n = 3 and x = 11, y = 5, n = 3, and claims in [6] to have dealt with the case C = 5. The case C = -1 was solved by Chao Ko, and an account appears in [3], pp. 302–304.

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Kolyvagin, V. A. "Fermat's equation over the tower of cyclotomic fields." Izvestiya: Mathematics 65, no. 3 (June 30, 2001): 503–41. http://dx.doi.org/10.1070/im2001v065n03abeh000337.

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Le, Maouha, and Ching Li. "On Fermat's equation in integral 2×2 matrices." Periodica Mathematica Hungarica 31, no. 3 (December 1995): 219–22. http://dx.doi.org/10.1007/bf01882197.

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Eynden, Charles Vanden. "Fermat's Last Theorem: 1637—1988." Mathematics Teacher 82, no. 8 (November 1989): 637–40. http://dx.doi.org/10.5951/mt.82.8.0637.

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Around 1637 the French jurist and amateur mathematician Pierre de Fermat wrote in the margin of his copy of Diophantus's Arithmetic that he had a “truly marvelous” proof that the equation xn + yn = zn has no solution in positive integers if n > 2. Unfortunately the margin was too narrow to contain it. In 1988 the world thought that the Japanese mathematician Yoichi Miyaoka, working at the Max Planck Institute in Bonn, West Germany, might have discovered a proof of this theorem. Such a proof would be of considerable interest because no evidence has been found that Fermat ever wrote one down, and no one has been able to find one in the 350 years since. In fact Miyaoka's announcement turned out to be premature, and a few weeks later articles reported holes in his argument that could not be repaired.

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Joseph, James E. "ALGEBRAIC PROOFS FERMAT'S LAST THEOREM, BEAL'S CONJECTURE." JOURNAL OF ADVANCES IN MATHEMATICS 12, no. 9 (September 27, 2016): 6576–77. http://dx.doi.org/10.24297/jam.v12i9.130.

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In this paper, the following statememt of Fermat's Last Theorem is proved. If x; y; z are positive integers, _ is an odd prime and z_ = x_ + y_; then x; y; z are all even. Also, in this paper, is proved Beal's conjecture; the equation z_ = x_ + y_ has no solution in relatively prime positive integers x; y; z; with _; _; _ primes at least 3:

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Ribenboim, Paulo. "Fermat's equation for matrices or quaternions over q-adic fields." Acta Arithmetica 113, no. 3 (2004): 241–50. http://dx.doi.org/10.4064/aa113-3-2.

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Mestechkin, M. "On periodic continued fractions, Pell equation, and Fermat's challenge numbers." Journal of Computational Methods in Sciences and Engineering 10, no. 1-2 (November 19, 2010): 49–66. http://dx.doi.org/10.3233/jcm-2010-0261.

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Dissertations / Theses on the topic "Fermat's equation"

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Esmonde, Jody. "Parametric solutions to the generalized Fermat equation." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape9/PQDD_0027/MQ50765.pdf.

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Meekin, Paul. "The Fermat equation over totally real fields." Thesis, University of Sheffield, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.397498.

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Deconinck, Heline. "The generalized Fermat equation over totally real number fields." Thesis, University of Warwick, 2016. http://wrap.warwick.ac.uk/81893/.

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Soto, Ballesteros Eduardo. "New results on modular forms and Fermat-type equations." Doctoral thesis, Universitat de Barcelona, 2019. http://hdl.handle.net/10803/667974.

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Aquesta tesi té dos objectius. El primer és contribuir en la teoria de congruències entre formes modulars via representacions de Galois. El segon és resoldre nous problemes diofantins de tipus Fermat. En quant a representacions de Galois considerem, per una banda, congruències entre formes modulars amb signes diferents en un primer Steinberg comú i, per l'altra, la pujada de nivell en primers arbitraris sota la condició (AbsIrr). Aquest últim treball té aplicacions en la construcció de cadenes segures que permeten propagar propietats com la modularitat. En quant a problemes diofantins, utilitzem les representacions de Galois i el ja clàssic mètode Modular per mostrar noves famílies de l'anomenada conjectura de Fermat Asimptòtica amb Coeficients.

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Barroso, de Freitas Nuno Ricardo. "Some Generalized Fermat-type Equations via Q-Curves and Modularity." Doctoral thesis, Universitat de Barcelona, 2012. http://hdl.handle.net/10803/91288.

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The main purpose of this thesis is to apply the modular approach to Diophantine equations to study some Fermat-type equations of signature (r; r; p) with r >/= 5 a fixed prime and “p” varying. In particular, we will study equations of the form x(r) + y(r) = Cz(p), where C is an integer divisible only by primes “q” is non-identical to 1; 0 (mod “r”) and obtain explicit arithmetic results for “r” = 5, 7, 13. We start with equations of the form x(5) + y(5) = Cz(p). Firstly, we attach two Frey curves E; F defined over Q(square root 5) to putative solutions of the equation. Then by using the work of J. Quer on embedding problems and on abelian varieties attached to Q-curves we prove that the p-adic Galois representations attached to E, F can be extended to p-adic representations E), (F) of Gal(Q=Q). Finally, we apply Serre's conjecture to the residual representations  (E), (F) and using Siksek's multi-Frey technique we conclude that the initial solution can not exist. We also describe a general method for attacking infinitely many equations of the form x(r) + y(r) = Cz(p) for all r>/= 7. The method makes use of elliptic curves over totally real fields, modularity and irreducibility results for representations attached to elliptic curves and level lowering theorems for Hilbert modular forms. Indeed, for each fixed “r” we produce several Frey curves defined over K+, the maximal totally real subfield of Q(xi-r). Moreover, if “r” is of the form 6k + 1 we prove the existence of a Frey curve defined over K(0) the subfield of K(+) of degree k. We prove also an irreducibility result for the mod “p” representations attached to certain elliptic curves and a modularity statement for elliptic curves over totally real abelian number fields satisfying some local conditions at 3. Finally, for r = 7 and r = 13 we are able to compute the required spaces of (Hilbert) newforms and by applying our general methods we obtain explicit arithmetic results for equations of signature (7; 7; p) and (13; 13; p). We end by providing two more Frey k-curves (a generalization of Q-curve), where “k” is a certain subfield of K(+), when “r” is a fixed prime of the form 4m+1.
En esta tesis, utilizaremos el método modular para profundizar en el estudio de las ecuaciones de tipo (r; r; p) para r un primo fijado. Empezamos por utilizar la teoría de J. Quer sobre variedades abelianas asociadas con Q-curvas y embedding problems para producir dos curvas de Frey asociadas con hipotéticas soluciones de infinitas ecuaciones de tipo (5; 5; p). Después, utilizando la conjetura de Serre y el método multi-Frey de Siksek demostraremos que las hipotéticas soluciones no pueden existir. Describiremos también un método general que nos permite atacar un número infinito de ecuaciones de tipo (r; r; p) para cada primo “r” mayor o igual que 7. El método hace uso de curvas elípticas sobre cuerpos de números, teoremas de modularidad, teoremas de bajada de nivel y formas modulares de Hilbert. Además, para ecuaciones de tipo (7; 7; p) y (13; 13; p) calcularemos los espacios de formas modulares relevantes y demostraremos que una familia infinita de ecuaciones no admite cierto tipo de soluciones. Además, demostraremos un nuevo teorema de modularidad para curvas elípticas sobre cuerpos totalmente reales abelianos. Finalmente, para primos congruentes con 1 módulo 4 propondremos dos curvas de Frey más. Demostraremos que son “k-curves” (una generalización de Q-curva) y también que satisfacen las propiedades necesarias para que pueda ser útiles en la aplicación del método modular.

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Silva, Filardes de Jesus Freitas da. "Equações diofantinas classicas e aplicações." [s.n.], 2009. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307049.

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Orientador: Emerson Alexandre de Oliveira Lima
Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Neste trabalho focalizamos os principais conceitos da teoria elementar dos números objetivando uma melhor compreensão das Equações Diofantinas Clássicas e suas aplicações e para isto explicitamos os conceitos de Números primos, Algoritmo de Euclides, Máximo divisor comum e Mínimo múltiplo comum, assim como a teoria das Congruências, uma abordagem sobre a Criptografica RSA e Soma de Inteiros. Palavras-Chave: Congruências Lineares, Soma de Inteiros, Equação de Fermat, Soma de Quadrados
Abstract: In this work we focus the main concepts of the elementary theory of numbers seeking a better understanding of Classical diophantine equations and their applications for this and explained the concepts of prime numbers, algorithms of Euclid, maximum common divisor and least common multiple and the theory of congruence , an approach on the RSA encryption and Sum of Integers. Keywords: Linear congruence, Sum of Integers, equation of Fermat, Sum of Squares
Mestrado
Teoria dos Numeros
Mestre em Matemática

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Alves, Lucinda Freese. "Aplicações de equações Diofantinas e um passeio pelo último teorema de Fermat." Universidade Federal de Goiás, 2017. http://repositorio.bc.ufg.br/tede/handle/tede/8104.

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The presente work aims to help students, teachers and lovers of mathematics, to better understand, interpret and solve problems that can be solved through Diophantine Equations. In this way, we present some basic concepts about Diophantine Equations as well as some practical applications. We also discuss Fermat ́s Last Theorem for the cases of n=2, n=3 and n=4, aiming to arouse interest, on the students, in Number Theory.
O presente trabalho tem como objetivo auxiliar estudantes, professores e apaixonados pela matemática, a melhor compreender, interpretar e resolver problemas que possam ser solucionados através das Equações Diofantinas. Desta forma, apresentamos alguns conceitos básicos sobre Equações Diofantinas bem como algumas aplicações práticas. Discutimos ainda, o Último Teorema de Fermat para os casos de n=2, n=3 e n=4, visando despertar o interesse no aluno pela teoria dos números.

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Nascimento, NatÃlia Medeiros do. "EquaÃÃes diofantinas e o mÃtodo das secantes e tangentes de Fermat." Universidade Federal do CearÃ, 2014. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=12409.

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CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior
Ao longo das Ãltimas dÃcadas, a transmissÃo do conhecimento matemÃtico na EducaÃÃo BÃsica sofreu diversas mudanÃas. âO Ensino Tradicionalâ da matemÃtica era baseado na memorizaÃÃo de fÃrmulas, havendo assim uma mecanizaÃÃo no processo de resoluÃÃo de problemas, onde o discente era visto como um ser passivo. A nova visÃo de ensino, que busca significar o que conteÃdo exposto em sala, motivou a escolha desse tema, visto que situaÃÃes problemas envolvendo equaÃÃes diofantinas podem ser facilmente percebidas em nosso cotidiano. O objetivo deste trabalho Ã oportunizar a realizaÃÃo de uma leitura consultiva para o professor do Ensino BÃsico, e asseverar que essas equaÃÃes podem ser aplicadas na EducaÃÃo BÃsica como uma ferramenta que instiga o pensamento lÃgico, o raciocÃnio, a compreensÃo e a interpretaÃÃo matemÃtica. A formulaÃÃo desse material que estÃ dividido em cinco capÃtulos se deu atravÃs de levantamento bibliogrÃfico por meio de pesquisas descritivas. A introduÃÃo compÃe o primeiro capÃtulo. O segundo capÃtulo versa sobre o Legado de Diofanto: vida e obras, ressaltando sua obra titulada âArithmeticaâ que contribuiu significativamente para o desenvolvimento da teoria dos nÃmeros. O terceiro capÃtulo trata das equaÃÃes diofantinas lineares de n variÃveis. O quarto capÃtulo aborda as ternas itagÃricas, o MÃtodo das Secantes e Tangentes de Fermat na busca de soluÃÃes racionais para quaÃÃes, com coeficientes racionais, da forma ax2+by2 = c, e um caso particular do Ãltimo Teorema de Fermat. O quinto capÃtulo Ã composto de problemas sobre equaÃÃes diofantinas lineares.
Over the past decades, the transmission of mathematical knowledge in basic education has undergone several changes. The âTeaching Traditionalâ math was based on memorizing formulas, so there mechanization in problem solving where the student was seen as a liability to be process. The new vision of education that seeks to signify exposed to room content, motivated the choice of this theme, as diophantine equations involving situations problems can be easily noticed in our daily lives. The objective of this work is an opportunity for a realization of an advisory reading for the teacher of basic education, and assert that these equations can be applied in basic education as a tool that encourages the logical thinking, reasoning, understanding and mathematical interpretation. The formulation of this material which is divided into five chapters was through literature review through descriptive research. The introduction comprises the first chapter. The second chapter deals with the Legacy of Diophantus: life and works, emphasizing his work entitled âArithmeticaâ which contributed significantly to the development of number theory. The third chapter deals with linear Diophantine equations in n variables. The fourth chapter discusses the Pythagorean tender, Fermatâs of secants and Tangents method, in finding rational solutions to equations with rational coefficients, of the form ax2 + by2 = c and a particular case Fermatâs Last Theorem. The fifth chapter is composed of problems on linear diophantine equations.

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Melo, Rômulo de Oliveira Lins Vieira de. "O método de circulantes, as fórmulas de Cardano e o teorema de Fermat para n=3." Universidade Federal da Paraíba, 2017. http://tede.biblioteca.ufpb.br:8080/handle/tede/9835.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
In this present work, principles and theorems associated to integers are returned, as well as eigenvalues and eigenvectors problems, highlighting a Hermitian matrix. Then it is emphasized to the Circulating Matrices, through which it is found the association to two well-defined polynomials: the representative and the characteristic. Later a brief account about the history of polynomial equations is made, drafting the Cardano-Tartaglia Formulas associated to them. Afterwards a unification is made in the resolution process of the polynomial equations of smaller degrees than the equal to 4, by means of the circulating matrices. The work is completed by proving a Fermat theorem for n = 3, using the Cardano-Tartaglia Formulas.
No presente trabalho, princípios e teoremas associados aos números inteiros são retomados, bem como problemas de autovalores e autovetores, sendo ressaltada a matriz Hermitiana. Em seguida é dado ênfase às Matrizes Circulantes, através das quais verifica-se a associação a dois polinômios bem definidos: o representante e o característico. Posteriormente realiza-se um breve relato acerca da história das equações polinomiais, destacandose as Fórmulas de Cardano-Tartaglia associadas às mesmas. Logo após é feita uma unificação no processo de resolução das equações polinomiais de graus menores do que o igual a 4, por meio das matrizes circulantes. O trabalho é finalizado, sendo provado o Teorema de Fermat para n = 3, recorrendo-se às Fórmulas de Cardano-Tartaglia.

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CAMPOS, Danilo Albuquerque de. "Algoritmos de aproximação de raízes quadradas." Universidade Federal Rural de Pernambuco, 2014. http://www.tede2.ufrpe.br:8080/tede2/handle/tede2/6699.

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In this work we are interested in showing three algorithms rational approximation of square roots by methods unknown or underutilized by teachers of elementary and secondary education. We begin by defining numerical sequence and convergence of sequences, will discuss the need to expand the concept of rational number and demonstrate the irrationality of the diagonal of a square. Prove an important theorem known in the literature as Dirichlet’s theorem and finally elencaremos three methods of approximating the square roots of natural non-perfect square numbers, very simple to be worked on in the classroom that are rational algorithm aproximção of Hiero of Alexandria, Theon’s Ladder and the Pell-Fermat equation, sende latter discursão fundamental to who will perform on the relationship of the three methods presented.
Neste trabalho estamos interessados em mostrar três algoritmos de aproximação racional de raízes quadradas por métodos pouco utilizados ou desconhecidos pelos professores do ensino fundamental e médio. Iniciaremos definindo sequência numérica e convergência de sequências, discutiremos sobre a necessidade de ampliação do conceito de número racional e demonstraremos a irracionalidade da diagonal de um quadrado. Provaremos um importante Teorema conhecido na literatura como o Teorema de Dirichlet, e por fim elencaremos três métodos de aproximação de raízes quadradas de números naturais não quadrados perfeitos, muito simples de serem trabalhados em sala de aula que são: O algoritmo de aproximação racional de Hierão de Alexandria, A escada de Theon e a Equação de Pell-Fermat, sendo este último fundamental para discussão que iremos realizar sobre a relação dos três métodos apresentados.

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Books on the topic "Fermat's equation"

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Fermat's last theorem. Providence, Rhode Island: American Mathematical Society, 2013.

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Inkeri, Kustaa. Collected papers of Kustaa Inkeri. Edited by Metsänkylä Tauno and Ribenboim Paulo. Kingston, Ont: Queen's University, 1992.

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Forms of Fermat equations and their zeta functions. Singapore: World Scientific, 2005.

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Tauno, Metsänkylä, and Ribenboim Paulo, eds. Collected papers of Kustaa Inkeri. Kingston, Ont: Queens's University, 1992.

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Apiosi︠a︡n, Levon Anik. Elementaren podkhod pri reshavane na vid neopredeleni uravnenii︠a︡ v t︠s︡eli chisla: Nauchno izsledvane s prakticheska nasochenost, pomagalo po matematika za naprednali. Plovdiv: Narodna biblioteka "Ivan Vazov", 2011.

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Ghandehari, Mostafa. Ray optics on surfaces. Arlington: Dept. of Mathematics, University of Texas at Arlington, 1997.

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Series, Michigan Historical Reprint. Proof of Fermat's theorem, and McGinnis' theorem of derivative equations in an absolute proof of Fermat's theorem; reduction of the general equation of ... supplementary theorems, by Michael Angelo Mc. Scholarly Publishing Office, University of Michigan Library, 2005.

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Coopersmith, Jennifer. The Lazy Universe. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198743040.001.0001.

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Action and the Principle of Least Action are explained: what Action is, why the Principle of Least Action works, why it underlies all physics, and what are the insights gained into energy, space, and time. The physical and mathematical origins of the Lagrange Equations, Hamilton’s Equations, the Lagrangian, the Hamiltonian, and the Hamilton-Jacobi Equation are shown. Also, worked examples in Lagrangian and Hamiltonian Mechanics are given. However the aim is to explain physics rather than to give a technical mastery of the subject. Therefore, much of the mathematics is in the appendices. While there is still some mathematics in the main text, the reader may select whether to work through, skim-read, or skip over it: the “story-line” will just about be maintained whatever route is chosen. The work is a much-reduced and simplified version of the outstanding text, “The Variational Principles of Mechanics” written by Cornelius Lanczos in 1949. That work is barely known today, and the present work may be considered as a tiny stepping-stone toward it. A principle that underlies all of physics will have wider repercussions; it is also to be appreciated in an aesthetic sense. It is hoped that this book will lead the reader to the widest possible understanding of the Principle of Least Action. Ideas such as Variational Mechanics, phase space, Fermat’s Principle, and Noether’s Theorem are explained.

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Brunjes, Lars. Forms Of Fermat Equations And Their Zeta Functions. World Scientific Publishing Company, 2004.

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Perlick, Volker. Ray Optics, Fermat's Principle, and Applications to General Relativity. Springer, 2000.

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Book chapters on the topic "Fermat's equation"

1

Bennett, Michael, Preda Mihăilescu, and Samir Siksek. "The Generalized Fermat Equation." In Open Problems in Mathematics, 173–205. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-32162-2_3.

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Harborth, Heiko. "Fermat-Like Binomial Equations." In Applications of Fibonacci Numbers, 1–5. Dordrecht: Springer Netherlands, 1988. http://dx.doi.org/10.1007/978-94-015-7801-1_1.

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Křížek, Michal, Florian Luca, and Lawrence Somer. "Fermat Primes and a Diophantine Equation." In 17 Lectures on Fermat Numbers, 117–29. New York, NY: Springer New York, 2002. http://dx.doi.org/10.1007/978-0-387-21850-2_11.

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Jarvis, Frazer. "Cyclotomic Fields and the Fermat Equation." In Springer Undergraduate Mathematics Series, 191–206. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-07545-7_9.

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Frey, Gerhard. "On Ternary Equations of Fermat Type and Relations with Elliptic Curves." In Modular Forms and Fermat’s Last Theorem, 527–48. New York, NY: Springer New York, 1997. http://dx.doi.org/10.1007/978-1-4612-1974-3_20.

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Darmon, Henri, and Claude Levesque. "Infinite Sums, Diophantine Equations and Fermat’s Last Theorem." In Developments in Mathematics, 73–95. Boston, MA: Springer US, 2002. http://dx.doi.org/10.1007/978-1-4757-3675-5_6.

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Pethó, A., E. Herrmann, and H. G. Zimmer. "S-integral points on elliptic curves and Fermat's triple equations." In Lecture Notes in Computer Science, 528–40. Berlin, Heidelberg: Springer Berlin Heidelberg, 1998. http://dx.doi.org/10.1007/bfb0054890.

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Li, Bao Qin. "On Fermat-Type Functional and Partial Differential Equations." In The Mathematical Legacy of Leon Ehrenpreis, 209–22. Milano: Springer Milan, 2012. http://dx.doi.org/10.1007/978-88-470-1947-8_13.

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Herrin, Judith. "Mathematical Mysteries in Byzantium." In Margins and Metropolis. Princeton University Press, 2013. http://dx.doi.org/10.23943/princeton/9780691153018.003.0015.

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Abstract:

This chapter examines how the mathematical mysteries of Diophantus were preserved, embellished, developed, and enjoyed in Byzantium by many generations of amateur mathematicians like Pierre de Fermat, who formulated what became known as Fermat's last theorem. Fermat was a seventeenth-century scholar and an amateur mathematician who developed several original concepts in addition to the famous “last theorem.” One of his sources was the Arithmetika, a collection of number problems written by Diophantus, a mathematician who appears to have flurished in Alexandria in the third century AD. It was through the Greek text translated into Latin that Fermat became familiar with Diophantus's mathematical problems, and in particular the one at book II, 8, which encouraged the formulation of his own last theorem. Fermat's last theorem claims that “the equation xn + yn = zn has no nontrivial solutions when n is greater than 2”.

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Wilson, Robin. "5. More triangles and squares." In Number Theory: A Very Short Introduction, 79–96. Oxford University Press, 2020. http://dx.doi.org/10.1093/actrade/9780198798095.003.0005.

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‘More triangles and squares’ explores Diophantine equations, named after the mathematician Diophantus of Alexandria. These are equations requiring whole number solutions. Which numbers can be written as the sum of two perfect squares? Joseph-Louis Lagrange’s theorem guarantees that every number can be written as the sum of four squares, and Edward Waring correctly suggested that there are similar results for higher powers. In 1637, Fermat conjectured that no three positive integers, a, b, and c, can satisfy the equation an+bn=cn, if n is greater than 2. Known as ‘Fermat’s last theorem’, this conjecture was eventually proved by Andrew Wiles in 1995.

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Conference papers on the topic "Fermat's equation"

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Jua´rez-Robles, D., A. Herna´ndez-Guerrero, C. E. Damia´n-Ascencio, and C. Rubio-Arana. "Three Dimensional Analysis of a PEM Fuel Cell With the Shape of a Fermat Spiral for the Flow Channel Configuration." In ASME 2008 International Mechanical Engineering Congress and Exposition. ASMEDC, 2008. http://dx.doi.org/10.1115/imece2008-68101.

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This work presents the analysis of a non-isothermal three-dimensional model in single phase of a PEM fuel cell with an innovative flow field path in the form of the Fermat spiral, i.e. two concentric spirals. The model is used to predict the current density contours and the water content in all of the zones of the fuel cell. The three-dimensional model includes: the gas flow channels with the shape of the new geometry proposed, the current collectors, gas diffusion layers, catalyst layers on both sides of the model, anode and cathode, and a proton exchange membrane in between. The model solves the energy equation, mass conservation, and species transport equations, including the source terms due the electrochemical effects occurring in the cell. The results show a higher average current density than the fuel cells with conventional flow paths, showing also that the current density attained is more uniform from the inlet to the outlet of the flow channels.

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