Number Theory

   

Cálculo de la Cantidad de Números Primos Que Hay Por Debajo de un Número Dado //// How to Calculate the Amount of Prime Numbers that Are Less Than a Given Number

Authors: Germán Paz

En este documento se explica un procedimiento para calcular la cantidad exacta de números primos que hay por debajo de cualquier número entero mayor que 4. Al momento de escribir este documento, se pensó que se había llegado a un resultado nuevo. Es por eso que este trabajo está escrito describiendo el mencionado procedimiento como algo nuevo. Sin embargo, mucho después de haber escrito este trabajo, se le señaló al autor que el procedimiento en cuestión ya había sido descubierto por Adrien-Marie Legendre.

De todos modos, este documento es útil para quien quiera entender el algoritmo de Legendre para calcular la cantidad exacta de números primos que hay por debajo de un número dado.

Para una demostración de las dos primeras fórmulas que aparecen en la pág. 72 (las cuales describen una de las propiedades del Triángulo de Pascal), ver vixra.org/abs/1303.0163 (Lemas 3 y 6). Combinando lo explicado en la pág. 39 y en las págs. 60 a 72 con las mencionadas fórmulas de la pág. 72, se puede ver la relación que existe entre una de las propiedades del Triángulo de Pascal y el algoritmo de Legendre.

Éste es un trabajo muy antiguo. En algunos casos se usó una simbología distinta a la convencional.

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In this paper we explain a procedure to calculate the exact amount of prime numbers that are less than an integer greater than 4. At the time of writing this paper, the author thought he had obtained a new result. This is why in this work the mentioned procedure is described as something new. However, a long time after this work had been written, it was pointed out to its author that the method described had already been discovered by Adrien-Marie Legendre.

Anyway, this paper is useful for anyone who wants to understand Legendre's algorithm to calculate the exact amount of primes that are less than a given number.

For a proof of the first two formulas that appear on page 72, see vixra.org/abs/1303.0163. If we combine what is explained on page 39 and on pages 60 to 72 with the mentioned formulas on page 72, we will see that there exists a relation between one of the properties of Pascal's Triangle and Legendre's algorithm.

This is a very old work.

Comments: 92 Pages. En español. // In Spanish.

Download: PDF

Submission history

[v1] 2013-06-09 20:56:07

Unique-IP document downloads: 21614 times

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