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Previous issue date: 2011-10-26 / The present investigation includes a study of Leonhard Euler and the pentagonal
numbers is his article Mirabilibus Proprietatibus Numerorum Pentagonalium - E524.
After a brief review of the life and work of Euler, we analyze the mathematical
concepts covered in that article as well as its historical context. For this purpose, we
explain the concept of figurate numbers, showing its mode of generation, as well as
its geometric and algebraic representations. Then, we present a brief history of the
search for the Eulerian pentagonal number theorem, based on his correspondence
on the subject with Daniel Bernoulli, Nikolaus Bernoulli, Christian Goldbach and Jean
Le Rond d'Alembert. At first, Euler states the theorem, but admits that he doesn t
know to prove it. Finally, in a letter to Goldbach in 1750, he presents a
demonstration, which is published in E541, along with an alternative proof. The
expansion of the concept of pentagonal number is then explained and justified by
compare the geometric and algebraic representations of the new pentagonal
numbers pentagonal numbers with those of traditional pentagonal numbers. Then we
explain to the pentagonal number theorem, that is, the fact that the infinite product(1
x)(1 xx)(1 x3)(1 x4)(1 x5)(1 x6)(1 x7)... is equal to the infinite series 1 x1 x2+x5+x7
x12 x15+x22+x26 ..., where the exponents are given by the pentagonal numbers
(expanded) and the sign is determined by whether as more or less as the exponent is
pentagonal number (traditional or expanded). We also mention that Euler relates the
pentagonal number theorem to other parts of mathematics, such as the concept of
partitions, generating functions, the theory of infinite products and the sum of
divisors. We end with an explanation of Euler s demonstration pentagonal number
theorem / O presente trabalho de pesquisa compreende em um estudo de Leonhard Euler
sobre os n?meros pentagonais e o artigo Mirabilibus Proprietatibus Numerorum
Pentagonalium -E524. Depois de uma breve revis?o da vida e obra de Euler,
analisamos os conceitos matem?ticos abordados no referido artigo como tamb?m a
sua contextualiza??o hist?rica. Para tanto, explicamos o conceito de n?meros
figurados, mostrando seu modo de gera??o, bem como suas representa??es
geom?tricas e alg?bricas. Em seguida, faz-se um pequeno hist?rico da busca
euleriana para o Teorema dos N?meros Pentagonais, perpassando sua
correspond?ncia sobre o assunto com Daniel Bernoulli, Nikolaus Bernoulli e
Christian Goldbach. No in?cio, Euler afirma o teorema, por?m admite que n?o sabe
demonstr?-lo. Finalmente, em uma carta ? Goldbach, de 1750, faz a procurada
demonstra??o, a qual ? publicada em E541, junto ? demonstra??o alternativa. A
expans?o do conceito de n?mero pentagonal ? ent?o explicada e justificada, tendo
em vista a compara??o das representa??es geom?trica e alg?brica dos novos
n?meros pentagonais com as dos n?meros pentagonais tradicionais. Em seguida,
explana-se o Teorema dos N?meros Pentagonais, isto ?, o fato de que o produto
infinito (1 x)(1 xx)(1 x 3)(1 x 4)(1 x 5)(1 x 6)(1 x 7) ... ser igual ? s?rie infinita 1 x 1
x 2+x 5+x 7 x 12 x 15+x 22+x 26 ..., onde os expoentes s?o dados pelos n?meros
pentagonais (expandidos) e o sinal ? dado como mais ou menos conforme o
expoente ? um n?mero pentagonal (seja tradicional, seja expandido) de ordem par
ou ?mpar. Tamb?m mencionamos que Euler, utiliza os n?meros pentagonais e o
referido teorema sobre outras partes da matem?tica, como: o conceito de parti??o,
fun??es geradoras, a teoria do produto infinito e a soma de divisores. Finalizamos
com uma explica??o da demonstra??o do Teorema dos N?meros Pentagonais.
| Identifer | oai:union.ndltd.org:IBICT/oai:repositorio.ufrn.br:123456789/16078 |
| Date | 26 October 2011 |
| Creators | Cota, Andreia Caroline da Silva |
| Contributors | CPF:13056476453, http://lattes.cnpq.br/2466525106349625, Morey, Bernadete Barbosa, CPF:59616571834, http://lattes.cnpq.br/7554818862651491, Baroni, Rosa Lucia Sverzut, CPF:83235191887, http://lattes.cnpq.br/3641041943819764, Fossa, John Andrew |
| Publisher | Universidade Federal do Rio Grande do Norte, Programa de P?s-Gradua??o em Ensino de Ci?ncias Naturais e Matem?tica, UFRN, BR, Ensino de Ci?ncias Naturais e Matem?tica |
| Source Sets | IBICT Brazilian ETDs |
| Language | Portuguese |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, info:eu-repo/semantics/masterThesis |
| Format | application/pdf |
| Source | reponame:Repositório Institucional da UFRN, instname:Universidade Federal do Rio Grande do Norte, instacron:UFRN |
| Rights | info:eu-repo/semantics/openAccess |
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