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The Factoradic IntegersBrinsfield, Joshua Sol 24 June 2016 (has links)
The arithmetic progressions under addition and composition satisfy the usual rules of arithmetic with a modified distributive law. The basic algebra of such mathematical structures is examined; this leads to the consideration of the integers as a metric space under the "factoradic metric", i.e., the integers equipped with a distance function defined by d(n,m)=1/N!, where N is the largest positive integer such that N! divides n-m. Via the process of metric completion, the integers are then extended to a larger set of numbers, the factoradic integers. The properties of the factoradic integers are developed in detail, with particular attention to prime factorization, exponentiation, infinite series, and continuous functions, as well as to polynomials and their extensions. The structure of the factoradic integers is highly dependent upon the distribution of the prime numbers and relates to various topics in algebra, number theory, and non-standard analysis. / Master of Science
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Uma confirmação da conjectura de Artin para pares de formas diagonais de graus 2 e 3Lelis, Jean Carlos Aguiar 10 November 2015 (has links)
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Previous issue date: 2015-11-10 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we present some methods used in the study of systems of additive forms
on local fields, and a proof for a particular case of Artin’s Conjecture, which says that
every systems with R additive forms of degrees k1; :::;kR has non trivial p-adic solution
for any prime p, if the number s of variables is higher than k2
1 +k2
2 + +k2R, given by
Wooley [12], where he shows that G(3;2) = 11.
Keywords / Nesse trabalho, nós apresentamos alguns dos métodos usados no estudo de formas
aditivas sobre corpos locais, e uma prova para um caso particular da Conjectura de
Artin, que afirma que todo sistema de R formas aditivas de graus k1;k2; :::;kR possui
solução p-ádica não trivial para todo p primo, se o número s de variáveis for maior que
k2
1 +k2
2 + +k2R
, dada por Wooley [12], onde ele mostra que G(3;2) = 11.
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