scholarly journals A Fibonacci-like Sequence of Composite Numbers

10.37236/1476 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
John W. Nicol
Keyword(s):  
Prime Numbers ◽  
Natural Numbers ◽  
Digit Pair

In 1964, Ronald Graham proved that there exist relatively prime natural numbers $a$ and $b$ such that the sequence $\{A_n\}$ defined by $$ {A}_{n} =A_{n-1}+A_{n-2}\qquad (n\ge 2;A_0=a,A_1=b)$$ contains no prime numbers, and constructed a 34-digit pair satisfying this condition. In 1990, Donald Knuth found a 17-digit pair satisfying the same conditions. That same year, noting an improvement to Knuth's computation, Herbert Wilf found a yet smaller 17-digit pair. Here we improve Graham's construction and generalize Wilf's note, and show that the 12-digit pair $$(a,b)= (407389224418,76343678551)$$ also defines such a sequence.

The diophantine equation x2 + paqb = yq

2021 ◽  
pp. 1-18
Author(s):  
Hemar Godinho ◽  
Victor G. L. Neumann
Keyword(s):  
Diophantine Equation ◽  
Prime Numbers ◽  
Natural Numbers ◽  
Lucas Sequences ◽  
Primitive Divisors

In this paper, we consider the Diophantine equation in the title, where [Formula: see text] are distinct odd prime numbers and [Formula: see text] are natural numbers. We present many results given conditions for the existence of integers solutions for this equation, according to the values of [Formula: see text] and [Formula: see text]. Our methods are elementary in nature and are based upon the study of the primitive divisors of certain Lucas sequences as well as the factorization of certain polynomials.

scholarly journals III. The sums of the series of the reciprocals of the prime numbers and of their powers

1882 ◽  
Vol 33 (216-219) ◽  
pp. 4-10 ◽  
Keyword(s):  
Prime Numbers ◽  
Natural Numbers ◽  
The Difference

Euler has shown that it is possible to sum the series of reciprocals of powers of the prime numbers, and he has calculated the values of these sums for the even powers. I thought it of some interest to calculate the sums for the odd powers, and to evaluate a peculiar constant (somewhat analogous to the Eulerian constant,— γ = 0·57721 56649 01532 86060 65) which presents itself, in the series of simple reciprocals of primes, as the difference between the sum of the series and the double logarithmic infinity to the Napierian base ϵ. The summation of these series was shown by Euler to depend upon the Napierian logarithms of the sums of the reciprocals of the powers of the natural numbers.

scholarly journals Some interesting tasks from the classical number theory

2020 ◽  
pp. 150-188
Author(s):  
Zurab Agdgomelashvili ◽  
Keyword(s):  
Number Theory ◽  
Prime Number ◽  
Prime Numbers ◽  
Natural Numbers ◽  
Prime Divisors ◽  
Problem Class

The article considers the following issues: – It’s of great interest for p and q primes to determine the number of those prime number divisors of a number 1 1 pq A p    that are less than p. With this purpose we have considered: Theorem 1. Let’s p and q are odd prime numbers and p  2q 1. Then from various individual divisors of the 1 1 pq A p    number, only one of them is less than p. A has at least two different simple divisors; Theorem 2. Let’s p and q are odd prime numbers and p  2q 1. Then all prime divisors of the number 1 1 pq A p    are greater than p; Theorem 3. Let’s q is an odd prime number, and p N \{1}, p]1;q] [q  2; 2q] , then each of the different prime divisors of the number 1 1 pq A p    taken separately is greater than p; Theorem 4. Let’s q is an odd prime number, and p{q1; 2q1}, then from different prime divisors of the number 1 1 pq A p    taken separately, only one of them is less than p. A has at least two different simple divisors. Task 1. Solve the equation 1 2 1 z x y y    in the natural numbers x , y, z. In addition, y must be a prime number. Task 2. Solve the equation 1 3 1 z x y y    in the natural numbers x , y, z. In addition, y must be a prime number. Task 3. Solve the equation 1 1 z x y p y    where p{6; 7; 11; 13;} are the prime numbers, x, y  N and y is a prime number. There is a lema with which the problem class can be easily solved: Lemma ●. Let’s a, b, nN and (a,b) 1. Let’s prove that if an  0 (mod| ab|) , or bn  0 (mod| ab|) , then | ab|1. Let’s solve the equations ( – ) in natural x , y numbers: I. 2 z x y z z x y          ; VI. (x  y)xy  x y ; II. (x  y)z  (2x)z  yz ; VII. (x  y)xy  yx ; III. (x  y)z  (3x)z  yz ; VIII. (x  y) y  (x  y)x , (x  y) ; IV. ( y  x)x y  x y , (y  x) ; IX. (x  y)x y  xxy ; V. ( y  x)x y  yx , (y  x) ; X. (x  y)xy  (x  y)x , (y  x) . Theorem . If a, bN (a,b) 1, then each of the divisors (a2  ab  b2 ) will be similar. The concept of pseudofibonacci numbers is introduced and some of their properties are found.

The sieve of Eratosthenes

1971 ◽  
Vol 18 (4) ◽  
pp. 236-237
Author(s):  
Roy Dubisch
Keyword(s):  
Prime Numbers ◽  
Fixed Number ◽  
Natural Numbers ◽  
Efficient Procedure

For some time I have been increasingly annoyed by what I regard as a serious flaw in the presentation in many texts of the celebrated sieve of Eratosthenes for obtaining prime numbers. Both the efficient procedure as conceived by Eratosthenes (see, for example, Boyer [1968], Eves [1953], or Heath [1921]) and the inefficient procedure frequently presented begin by listing ail the natural numbers up to some fixed number n. (I am aware that we Jist the numerals for the numbers, but I don't want to be bothered later by having to say “the number named by the numeral”!)

scholarly journals Introducing Two New Sieves for Factorization Natural Odd Numbers

2020 ◽  
Author(s):  
Ramazanali Maleki Chorei
Keyword(s):  
Factorization Method ◽  
Prime Numbers ◽  
Natural Numbers

For each non-prime odd number as F=pq , if we consider m/n as an approximation for q/p and choose k=mn , then by proving some lemmas and theorems, we can compute the values of m and n. Finally, by using Fermat’s factorization method for F and 4kF as difference of two non-consecutive natural numbers, we should be able to find the values of p and q. Then we introduce two new and powerful sieves for separating composite numbers from prime numbers.

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 A note on the congruent distribution of the number of prime factors of natural numbers

2001 ◽  
Vol 163 ◽  
pp. 1-11 ◽  
Author(s):  
Tomio Kubota ◽  
Mariko Yoshida
Keyword(s):  
Analytic Continuation ◽  
Natural Number ◽  
Dirichlet Series ◽  
Half Plane ◽  
Arithmetic Function ◽  
Prime Numbers ◽  
Natural Numbers ◽  
Prime Factors

Let n = p1p2 … pr be a product of r prime numbers which are not necessarily different. We define then an arithmetic function µm(n) bywhere m is a natural number. We further define the function L(s, µm) by the Dirichlet seriesand will show that L(s, µm), (m ≥ 3), has an infinitely many valued analytic continuation into the half plane Re s > ½.

scholarly journals Prime Numbers

2017 ◽  
Vol 5 (6) ◽  
pp. 41-66
Author(s):  
Mady Ndiaye
Keyword(s):  
Information Technology ◽  
Natural Number ◽  
Prime Number ◽  
Mathematical Reasoning ◽  
Prime Numbers ◽  
The Other ◽  
Natural Numbers ◽  
Mathematical Formulas ◽  
The Universe

A prime number is a natural number that has Just two divisors: one and itself. From antiquity until our time, scientists are researching mathematical reasoning to understand the prime numbers; eminent scholars had worked on this field before it is abandoned. Mathematicians considered the prime numbers like « building blocs in building natural numbers » and the field of mathematics the most difficult. Everything is about numbers, everything is about measure, The understanding of the natural numbers and more general the understanding of the numbers depend on the understanding of the prime numbers. This understanding of the prime will gives us greater ease to understand the other sciences. The prime numbers play a very important role for securing information technology hence promotion of the NTIC, Every year, there is a price for persons who will discover the biggest prime “it‟s the hunt for the big prime” This first part of this article about the prime numbers has taken a weight off the scientists „s shoulders by highlighting the universe of the prime numbers and has bring the problem of the prime numbers to an end. The mathematical formulas set out in this article allow us to determine all the biggest prime numbers compared to the capacity of our machines.

scholarly journals On the order of the non-primes of type 6n+5 and 6n+1

2021 ◽  
Author(s):  
Christoph Wolmersdorfer
Keyword(s):  
Random Distribution ◽  
Prime Numbers ◽  
Fundamental Importance ◽  
Natural Numbers ◽  
Unique Product ◽  
Primality Test ◽  
Sine Functions

Abstract The fundamental importance of prime numbers for mathematics is that all other natural numbers can be represented as a unique product of prime numbers[1]. For centuries, mathematicians have been trying to find an order in the occurrence of prime numbers. Since Riemann´s paper on the number of primes under a given size, the distribution of the primes is assumed to be random[2]. Here, I show that prime numbers are not randomly distributed. Their positions are determined by the order of occurrence of the nonprimes of types 6n+5 and 6n+1. By using parametric sine functions, I will show here that all assumptions known today regarding the distribution of prime numbers larger than 3 are wrong. In particular, this refutes Riemann´s hypothesis of random distribution of prime numbers. Furthermore, I will show an exact primality test based on these parametric sine functions, which only uses one parameter.

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