Decomposição de números fatoriais em fatores primos: algoritmos para situações complementares ao Teorema de Legendre
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Data
2022-04-05
Autores
Carvalho, Leonardo Henrique Melo de
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Editor
Universidade Estadual Paulista (Unesp)
Resumo
O principal objetivo desse trabalho é desenvolver um algoritmo baseado no Teorema de Legendre, que conta quantos fatores primos existem na decomposição do fatorial de um número natural n, bem como, descrever uma atividade que envolvesse Teoria dos Números e programação computacional básica, voltada para estudantes da educação básica. Detalhamos a construção dos objetos do Teorema de Legendre sobre o ponto de vista das duas funções envolvidas. Em seguida, descrevemos dois algoritmos que tratam situações complementares ao Teorema de Legendre, sendo um deles apresentado no livro “Aritmética” da coleção PROFMAT HEFEZ (2016), que teve sua base teórica explicitada detalhadamente, complementando o que foi apresentado na referência e outro desenvolvido de forma independente pelos autores do trabalho, com sua respectiva base teórica construída em detalhes. Por fim, foi construída uma atividade voltada aos estudantes de educação básica que integrasse aspectos da Teoria do Números, aprendidos durante essa fase de ensino, bem como, ideias de programação básica usando o “VisualG”.
The goal of this work is to develop an algorithm based on Legendre’s Theorem, which counts how many prime factors there are in the factorial decomposition of a natural number n, as well as to describe an activity that involves Number Theory and computer programming adressed for basic education students. We seek to understand the construction of Legendre’s Theorem objects from the point of view of the two functions involved. Next, we describe two algorithms that deal with situations that are complementary to Legendre’s Theorem, one of which presented in the book “Aritmética” HEFEZ (2016) of the PROFMAT collection, which had its theoretical basis explained,and a complementary one, independently developed by the authors, with the respective theoretical basis built in detail. Finally, the construction of an activity for basic education students was described, integrating aspects of Number Theory learned during this teaching phase with basic programming ideas using “VisualG”.
The goal of this work is to develop an algorithm based on Legendre’s Theorem, which counts how many prime factors there are in the factorial decomposition of a natural number n, as well as to describe an activity that involves Number Theory and computer programming adressed for basic education students. We seek to understand the construction of Legendre’s Theorem objects from the point of view of the two functions involved. Next, we describe two algorithms that deal with situations that are complementary to Legendre’s Theorem, one of which presented in the book “Aritmética” HEFEZ (2016) of the PROFMAT collection, which had its theoretical basis explained,and a complementary one, independently developed by the authors, with the respective theoretical basis built in detail. Finally, the construction of an activity for basic education students was described, integrating aspects of Number Theory learned during this teaching phase with basic programming ideas using “VisualG”.
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Palavras-chave
Legendre, Fatores primos, Teorema Fundamental da Aritmética, Aritmética, Teorema de Legendre, Prime factors, Fundamental Theorem of Arithimetic, Arithimetic, Theorem’s Legendre