3. Prime-time mathematics

2020 ◽  
pp. 37-57
Author(s):  
Robin Wilson
Keyword(s):  
Number Theory ◽  
Prime Number ◽  
Prime Numbers ◽  
Prime Time ◽  
Leonhard Euler ◽  
Mersenne Primes ◽  
Fermat Primes

‘Prime-time mathematics’ explores prime numbers, which lie at the heart of number theory. Some primes cluster together and some are widely spread, while primes go on forever. The Sieve of Eratosthenes (3rd century BC) is an ancient method for identifying primes by iteratively marking the multiples of each prime as not prime. Every integer greater than 1 is either a prime number or can be written as a product of primes. Mersenne primes, named after French friar Marin de Mersenne, are prime numbers that are one less than a power of 2. Pierre de Fermat and Leonhard Euler were also prime number enthusiasts. The five Fermat primes are used in a problem from geometry.

scholarly journals Considerations for Goldbach's Strong Conjecture

2021 ◽  
Author(s):  
Carleilton Severino Silva
Keyword(s):  
Number Theory ◽  
Prime Numbers ◽  
Weak Version ◽  
Strong Version ◽  
Leonhard Euler

Since 1742, the year in which the Prussian Christian Goldbach wrote a letter to Leonhard Euler with his Conjecture in the weak version, mathematicians have been working on the problem. The tools in number theory become the most sophisticated thanks to the resolution solutions. Euler himself said he was unable to prove it. The weak guess in the modern version states the following: any odd number greater than 5 can be written as the sum of 3 primes. In response to Goldbach's letter, Euler reminded him of a conversation in which he proposed what is now known as Goldbach's strong conjecture: any even number greater than 2 can be written as a sum of 2 prime numbers. The most interesting result came in 2013, with proof of weak version by the Peruvian Mathematician Harald Helfgott, however the strong version remained without a definitive proof. The weak version can be demonstrated without major difficulties and will not be described in this article, as it becomes a corollary of the strong version. Despite the enormous intellectual baggage that great mathematicians have had over the centuries, the Conjecture in question has not been validated or refuted until today.

scholarly journals THE QUOTIENT SET OF -GENERALISED FIBONACCI NUMBERS IS DENSE IN

2017 ◽  
Vol 96 (1) ◽  
pp. 24-29 ◽  
Author(s):  
CARLO SANNA
Keyword(s):  
Number Theory ◽  
Prime Number ◽  
Algebraic Number ◽  
Fibonacci Numbers ◽  
Prime Numbers ◽  
Algebraic Number Theory ◽  
Initial Values ◽  
Quotient Set ◽  
Successive Term

The quotient set of $A\subseteq \mathbb{N}$ is defined as $R(A):=\{a/b:a,b\in A,b\neq 0\}$. Using algebraic number theory in $\mathbb{Q}(\sqrt{5})$, Garcia and Luca [‘Quotients of Fibonacci numbers’, Amer. Math. Monthly, to appear] proved that the quotient set of Fibonacci numbers is dense in the $p$-adic numbers $\mathbb{Q}_{p}$ for all prime numbers $p$. For any integer $k\geq 2$, let $(F_{n}^{(k)})_{n\geq -(k-2)}$ be the sequence of $k$-generalised Fibonacci numbers, defined by the initial values $0,0,\ldots ,0,1$ ($k$ terms) and such that each successive term is the sum of the $k$ preceding terms. We use $p$-adic analysis to generalise the result of Garcia and Luca, by proving that the quotient set of $k$-generalised Fibonacci numbers is dense in $\mathbb{Q}_{p}$ for any integer $k\geq 2$ and any prime number $p$.

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.

scholarly journals Considerations for Goldbach's Strong Conjecture

2021 ◽  
Author(s):  
Carleilton Severino Silva
Keyword(s):  
Number Theory ◽  
Prime Numbers ◽  
Weak Version ◽  
Strong Version ◽  
Leonhard Euler

Since 1742, the year in which the Prussian Christian Goldbach wrote a letter to Leonhard Euler with his Conjecture in the weak version, mathematicians have been working on the problem. The tools in number theory become the most sophisticated thanks to the resolution solutions. Euler himself said he was unable to prove it. The weak guess in the modern version states the following: any odd number greater than 5 can be written as the sum of 3 primes. In response to Goldbach's letter, Euler reminded him of a conversation in which he proposed what is now known as Goldbach's strong conjecture: any even number greater than 2 can be written as a sum of 2 prime numbers. The most interesting result came in 2013, with proof of weak version by the Peruvian Mathematician Harald Helfgott, however the strong version remained without a definitive proof. The weak version can be demonstrated without major difficulties and will not be described in this article, as it becomes a corollary of the strong version. Despite the enormous intellectual baggage that great mathematicians have had over the centuries, the Conjecture in question has not been validated or refuted until today.

scholarly journals Some New Notes on Mersenne Primes and Perfect Numbers

2020 ◽  
Vol 3 (1) ◽  
pp. 15
Author(s):  
Leomarich F Casinillo
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Mathematical Structure ◽  
Prime Numbers ◽  
Perfect Number ◽  
Positive Divisor ◽  
Triangular Numbers ◽  
Perfect Numbers ◽  
Mersenne Primes ◽  
Mersenne Prime

<p>Mersenne primes are specific type of prime numbers that can be derived using the formula <img title="\large M_p=2^{p}-1" src="https://latex.codecogs.com/gif.latex?\large&amp;space;M_p=2^{p}-1" alt="" />, where <img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&amp;space;p" alt="" /> is a prime number. A perfect number is a positive integer of the form <img title="\large P(p)=2^{p-1}(2^{p}-1)" src="https://latex.codecogs.com/gif.latex?\large&amp;space;P(p)=2^{p-1}(2^{p}-1)" alt="" /> where <img title="\large 2^{p}-1" src="https://latex.codecogs.com/gif.latex?\large&amp;space;2^{p}-1" alt="" /> is prime and <img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&amp;space;p" alt="" /> is a Mersenne prime, and that can be written as the sum of its proper divisor, that is, a number that is half the sum of all of its positive divisor. In this note, some concepts relating to Mersenne primes and perfect numbers were revisited. Further, Mersenne primes and perfect numbers were evaluated using triangular numbers. This note also discussed how to partition perfect numbers into odd cubes for odd prime <img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&amp;space;p" alt="" />. Also, the formula that partition perfect numbers in terms of its proper divisors were constructed and determine the number of primes in the partition and discuss some concepts. The results of this study is useful to better understand the mathematical structure of Mersenne primes and perfect numbers.</p>

scholarly journals R-prime numbers of degree k

2018 ◽  
Vol 38 (2) ◽  
pp. 75-82
Author(s):  
Abdelhakim Chillali
Keyword(s):  
Number Theory ◽  
Prime Number ◽  
Complexity Theory ◽  
Prime Numbers ◽  
Prime Pair ◽  
Random Input ◽  
Trap Door ◽  
Twin Primes ◽  
Open Questions ◽  
One Way Function

In computer science, a one-way function is a function that is easy to compute on every input, but hard to invert given the image of a random input. Here, "easy" and "hard" are to be understood in the sense of computational complexity theory, specifically the theory of polynomial time problems. Not being one-to-one is not considered sufficient of a function for it to be called one-way (see Theoretical Definition, in article). A twin prime is a prime number that has a prime gap of two, in other words, differs from another prime number by two, for example the twin prime pair (5,3). The question of whether there exist infinitely many twin primes has been one of the great open questions in number theory for many years. This is the content of the twin prime conjecture, which states: There are infinitely many primes p such that p + 2 is also prime. In this work we define a new notion: ‘r-prime number of degree k’ and   we give a new RSA trap-door one-way. This notion generalized a twin prime numbers because the twin prime numbers are 2-prime numbers of degree 1.

scholarly journals CARACTERIZACIÓN DE NÚMEROS PRIMOS

2017 ◽  
Vol 22 (2) ◽  
pp. 25
Author(s):  
Héctor Carlos Guimaray Huerta
Keyword(s):  
Number Theory ◽  
Prime Number ◽  
Prime Numbers ◽  
Composite Number ◽  
Palabras Clave

Los números primos es motivo de investigación en la teoría de números; en la actualidad, no existe una fórmula que nos permita obtener dichos números, y que la distribución de los mismos se considera que es aleatoria. Lo que existe son métodos para averiguar si un número es primo o compuesto. En el presente artículo se presenta una caracterización de números primos que es el complemento de los números compuestos. Palabras clave.- Divisor, Número primo, Número compuesto, Caracterización, Conjetura. ABSTRACTThe prime numbers motivate the investigation in number theory; nowadays, does not exist a formula that allows get those numbers, and that the distribution thereof is considered random. There are methods to find whether a number is prime or composite. This article presents a characterization of prime numbers which is the complement of composite numbers. Keywords.- Divisor, Prime number, Composite number, Characterization, Conjecture.

Euler's contribution to number theory

2007 ◽  
Vol 91 (522) ◽  
pp. 453-461 ◽  
Author(s):  
Peter Shiu
Keyword(s):  
Eighteenth Century ◽  
Number Theory ◽  
Scientific Output ◽  
Scholarly Work ◽  
Long Period ◽  
Peter The Great ◽  
Complete Work ◽  
Leonhard Euler ◽  
The Subject ◽  
Historic Background

Individuals who excel in mathematics have always enjoyed a well deserved high reputation. Nevertheless, a few hundred years back, as an honourable occupation with means to social advancement, such an individual would need a patron in order to sustain the creative activities over a long period. Leonhard Euler (1707-1783) had the fortune of being supported successively by Peter the Great (1672-1725), Frederich the Great (1712-1786) and the Great Empress Catherine (1729-1791), enabling him to become the leading mathematician who dominated much of the eighteenth century. In this note celebrating his tercentenary, I shall mention his work in number theory which extended over some fifty years. Although it makes up only a small part of his immense scientific output (it occupies only four volumes out of more than seventy of his complete work) it is mostly through his research in number theory that he will be remembered as a mathematician, and it is clear that arithmetic gave him the most satisfaction and also much frustration. Gazette readers will be familiar with many of his results which are very well explained in H. Davenport's famous text [1], and those who want to know more about the historic background, together with the rest of the subject matter itself, should consult A. Weil's definitive scholarly work [2], on which much of what I write is based. Some of the topics being mentioned here are also set out in Euler's own Introductio in analysin infinitorum (1748), which has now been translated into English [3].

scholarly journals The continuity of prime numbers can lead to even continuity(The ultimate)

2021 ◽  
Author(s):  
Xie Ling
Keyword(s):  
Prime Number ◽  
Prime Numbers ◽  
Original Group

Abstract n continuous prime numbers can combine a group of continuous even numbers. If an adjacent prime number is followed, the even number will continue. For example, if we take prime number 3, we can get even number 6. If we follow an adjacent prime number 5, we can get even numbers by using 3 and 5: 6, 8 and 10. If a group of continuous prime numbers 3,5,7,11,... P, we can get a group of continuous even numbers 6,8,10,12,..., 2n. Then if an adjacent prime number q is followed, the original group of even numbers 6,8,10,12,..., 2n will be finitely extended to 2 (n + 1) or more adjacent even numbers. My purpose is to prove that the continuity of prime numbers will lead to even continuity as long as 2 (n + 1) can be extended. If the continuity of even numbers is discontinuous, it violates the Bertrand Chebyshev theorem of prime numbers. Because there are infinitely many prime numbers: 3, 5, 7, 11,... We can get infinitely many continuous even numbers: 6,8,10,12,...

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