scholarly journals Multiplication tables for non-prime odd numbers The untouched tables which will give us a better understanding of the distribution of prime odd numbers

2020 ◽  
Author(s):  
ahmad hazaymeh
Keyword(s):  
Prime Numbers ◽  
Sieve Method ◽  
Random Series ◽  
Multiplication Tables ◽  
Counter Example ◽  
Partial Multiplication

The sieve method is used to separate prime numbers from non-prime numbers. If the set of prime odd numbers cannot be written as multiplication tables, the set of non-prime odd numbers can be written as multiplication tables. Thus, each odd number that does not appear in these multiplication tables is certainly a prime odd number. Based on these tables, it was proved by the opposite method that the series of prime numbers are random series. Although they are random, they can be easily tracked using the opposite method. The counter-example is used to proof that it is not possible to write whole multiplication tables of prime odd numbers on formula of [(𝑎×𝑏)+𝑐] or [(𝑎×𝑏)−𝑐]. Instead, partial multiplication tables can be used. It also was proved that the number 1 is a prime odd number.

scholarly journals Números primos; método gráfico de la conjetura de GOLDBACH

2018 ◽  
Vol 3 (2) ◽  
pp. 77
Author(s):  
Yandry Marcelo Intriago Delgado
Keyword(s):  
Prime Numbers ◽  
Prime Element ◽  
Continuous Line ◽  
Natural Numbers ◽  
Microsoft Excel ◽  
Simple Method ◽  
Multiplication Tables ◽  
Palabras Clave ◽  
Combinatorial Formulas ◽  
Minimum Number

Mediante el uso de Microsoft Excel el siguiente trabajo examina las tablas de multiplicar desde una perspectiva distinta, con un método sencillo para encontrar la secuencia de los números primos en la línea continua de los números naturales , y así luego se identifican gráficamente los números que cumplen con la Conjetura de Goldbach, al realizar una triangulación con líneas que unen la series de los  y ; siendo la notación: el cuadrado de los números naturales. A continuación, se trazan diagonales paralelas a las sucesiones y únicamente en cada elemento primo  de la línea de los  y así se obtienen intersecciones que cumplen con la conjetura fuerte de Goldbach. Se aplican   fórmulas para calcular el número mínimo de intersecciones que se generan en un conjunto de los  consecutivos. Así mismo, para obtener la conjetura débil de Goldbach, se puede usar el gráfico ya antes mencionado, y se emplean fórmulas combinatorias. Este método permite identificar el intervalo de afectación que tiene un elemento primo en la secuencia de los naturales y modelar una línea continua, que revela un gráfico similar al que se conoce como cometa de Goldbach. Palabras clave: Gráfico, números primos, conjetura de Goldbach. ABSTRACT By the use of Microsoft Excel the following work examines the multiplication tables from a different perspective, with a simple method to find the sequence of the prime numbers in the continuous line of the natural numbers ( ), and then we can graphically identify the numbers that comply with the Goldbach Conjecture, when making a triangulation with lines that join the series of the  and , in this article the notation:  is the square of the natural numbers. Next, diagonals are drawn parallel to the sequence  and  only in each prime element  of the line of the  and thus intersections are obtained that meet the strong conjecture of Goldbach. Formulas are applied to calculate the minimum number of intersections that are generated in a set of consecutive . Likewise, to obtain the weak Goldbach conjecture, the aforementioned graph can be used, and combinatorial formulas are used. This method serves to identify the range of affectation that a prime element has in the sequence of the natural numbers, and to model a continuous line, which reveals a graph similar to what is known as Goldbach's comet. Key words: Graph, prime numbers, Goldbach conjecture.

scholarly journals The mathematical constraint of the counterexample of Goldbach's strong conjecture

2020 ◽  
Author(s):  
ahmad hazaymeh ◽  
Khaled Hazaymeh ◽  
Sukaina Hazaymeh
Keyword(s):  
Prime Number ◽  
Prime Number Theorem ◽  
Prime Numbers ◽  
Counter Example ◽  
Logical Conclusion

In this paper, we have demonstrated a proof that the Counterexample of Goldbach's strong conjecture is impossible in two steps: First, we reformulated Goldbach's strong conjecture using the subtraction connotation. Second: the mathematical Constraint that must be fulfilled in any even number has been deduced to be that even number a Counterexample of Goldbach's strong conjecture. Then we demonstrated that any counterexample would fulfill this mathematical Constraint. It will either contradict the theorem of infinite prime numbers or contradict the Prime Number Theorem. Therefore, the logical conclusion is that there is no counterexample to Goldbach's strong conjecture. With the absence of a counter-example, Goldbach's strong conjecture would be a true conjecture

Deciding Associativity for Partial Multiplication Tables of Order 3

1978 ◽  
Vol 32 (142) ◽  
pp. 593 ◽  
Author(s):  
Paul W. Bunting ◽  
Jan van Leeuwen ◽  
Dov Tamari
Keyword(s):  
Multiplication Tables ◽  
Partial Multiplication

Números primos; método gráfico de la conjetura de GOLDBACH

2018 ◽  
Vol 3 (2) ◽  
pp. 77
Author(s):  
Yandry Intriago Delgado
Keyword(s):  
Prime Numbers ◽  
Prime Element ◽  
Continuous Line ◽  
Natural Numbers ◽  
Microsoft Excel ◽  
Simple Method ◽  
Multiplication Tables ◽  
Palabras Clave ◽  
Combinatorial Formulas ◽  
Minimum Number

Mediante el uso de Microsoft Excel el siguiente trabajo examina las tablas de multiplicar desde una perspectiva distinta, con un método sencillo para encontrar la secuencia de los números primos en la línea continua de los números naturales , y así luego se identifican gráficamente los números que cumplen con la Conjetura de Goldbach, al realizar una triangulación con líneas que unen la series de los  y ; siendo la notación: el cuadrado de los números naturales. A continuación, se trazan diagonales paralelas a las sucesiones y únicamente en cada elemento primo  de la línea de los  y así se obtienen intersecciones que cumplen con la conjetura fuerte de Goldbach. Se aplican   fórmulas para calcular el número mínimo de intersecciones que se generan en un conjunto de los  consecutivos. Así mismo, para obtener la conjetura débil de Goldbach, se puede usar el gráfico ya antes mencionado, y se emplean fórmulas combinatorias. Este método permite identificar el intervalo de afectación que tiene un elemento primo en la secuencia de los naturales y modelar una línea continua, que revela un gráfico similar al que se conoce como cometa de Goldbach. Palabras clave: Gráfico, números primos, conjetura de Goldbach. ABSTRACT By the use of Microsoft Excel the following work examines the multiplication tables from a different perspective, with a simple method to find the sequence of the prime numbers in the continuous line of the natural numbers ( ), and then we can graphically identify the numbers that comply with the Goldbach Conjecture, when making a triangulation with lines that join the series of the  and , in this article the notation:  is the square of the natural numbers. Next, diagonals are drawn parallel to the sequence  and  only in each prime element  of the line of the  and thus intersections are obtained that meet the strong conjecture of Goldbach. Formulas are applied to calculate the minimum number of intersections that are generated in a set of consecutive . Likewise, to obtain the weak Goldbach conjecture, the aforementioned graph can be used, and combinatorial formulas are used. This method serves to identify the range of affectation that a prime element has in the sequence of the natural numbers, and to model a continuous line, which reveals a graph similar to what is known as Goldbach's comet. Key words: Graph, prime numbers, Goldbach conjecture.

scholarly journals Deciding associativity for partial multiplication tables of order $3$

1978 ◽  
Vol 32 (142) ◽  
pp. 593-593
Author(s):  
Paul W. Bunting ◽  
Jan van Leeuwen ◽  
Dov Tamari
Keyword(s):  
Multiplication Tables ◽  
Partial Multiplication

A remark on a theorem of the Goldbach-Waring type

2004 ◽  
Vol 41 (3) ◽  
pp. 309-324
Author(s):  
C. Bauer
Keyword(s):  
Prime Numbers

Let pi, 2 ≤ i ≤ 5 be prime numbers. It is proved that all but ≪ x23027/23040+ε even integers N ≤ x can be written as N = p21 + p32 + p43 + p45.

The First 50 Million Prime Numbers

1977 ◽  
Vol 1 (S2) ◽  
pp. 7-19 ◽  
Author(s):  
Don Zagier
Keyword(s):  
Prime Numbers

Density of summable subsequences of a sequence and its applications

2020 ◽  
Vol 70 (3) ◽  
pp. 657-666
Author(s):  
Bingzhe Hou ◽  
Yue Xin ◽  
Aihua Zhang
Keyword(s):  
Prime Numbers ◽  
Linear Operators ◽  
Upper Density ◽  
Asymptotic Densities

AbstractLet x = $\begin{array}{} \displaystyle \{x_n\}_{n=1}^{\infty} \end{array}$ be a sequence of positive numbers, and 𝓙x be the collection of all subsets A ⊆ ℕ such that $\begin{array}{} \displaystyle \sum_{k\in A} \end{array}$xk < +∞. The aim of this article is to study how large the summable subsequence could be. We define the upper density of summable subsequences of x as the supremum of the upper asymptotic densities over 𝓙x, SUD in brief, and we denote it by D*(x). Similarly, the lower density of summable subsequences of x is defined as the supremum of the lower asymptotic densities over 𝓙x, SLD in brief, and we denote it by D*(x). We study the properties of SUD and SLD, and also give some examples. One of our main results is that the SUD of a non-increasing sequence of positive numbers tending to zero is either 0 or 1. Furthermore, we obtain that for a non-increasing sequence, D*(x) = 1 if and only if $\begin{array}{} \displaystyle \liminf_{k\to\infty}nx_n=0, \end{array}$ which is an analogue of Cauchy condensation test. In particular, we prove that the SUD of the sequence of the reciprocals of all prime numbers is 1 and its SLD is 0. Moreover, we apply the results in this topic to improve some results for distributionally chaotic linear operators.

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