On the trapezoids of number of divisors, their relationship with Pascal's triangle and their use in counting problems

Abstract

This article was carried out to deal with the trapezoids of quantity of numbers dividers where the way of obtaining its elements is similar to Pascaline triangle, which allow to determine the number of dividers of the numerical forms. This relates to counting forms where there are repeated elements, aspects that would make it necessary to study new situations in combinatorial theory, which open up new possibilities for the analysis of ways to choose each number of elements of a certain collection of them. In the development of it, preliminary aspects were presented, on Gauss’s theorem and the Pascal triangle as well as certain simple theoretical aspects and then show the divider trapezoids with their definition, examples, characteristics, ways to generate them and their possible applications. It is necessary to clarify that a definitive theorem is not presented on the number of dividers, because it is impossible, since each trapezoid serves the nature of certain numbers and since each number has different representations (which is explained by the fundamental theorem of the arithmetic of Gauss), this requires building a particular trapezoid in each case. In addition, a part called questions and reflections such as Newton Queries in their Optical book are included, where a series of questions are made for analysis and searching more possible uses of this mathematical tool and its application in statistics. Lastly, in the conclusion, certain aspects are indicated in relation to the possible scope of the number of number dividers.