Numerical analysis approach to twin primes conjecture

The purpose of this paper is to demonstrate how the modified Sieve of Eratosthenes is used to have an approach to twin prime conjecture. If the Sieve is used in its basic form, it does not produce anything new. If it is used through the numerical analysis method explained in this paper, we obtain a specific counting of twin primes. This counting is based on the false assumption that distribution of primes follows punctually the Logarithmic Integral function denoted as Li(x) (see [5] and [10], pp. 174–176). It may be a starting point for future research based on this numerical analysis method technique that can be used in other mathematical branches.

Twin Prime Conjecture - Proof that there are infinite twin primes.

We determine that twin primes are not random within specific range of number series. Within the range, they are part of an infinite repeating cycle within the series. The repeating cycles of this series is symmetrical and that range is part of one such cycle. We can also determine where in the repeating cycle this range is located. Once we establish the repeating cycles, we find the probability of twin primes for the cycle/series. Next we determine the lower bound for the probability and the range. Finally, we prove that the twin prime conjecture is true.

NEW RESULTS ON K -PRIME LABELING

In this paper, we introduce new results on k-prime labeling. First, we discuss the kprime labeling of cycles Cn for some values of k and n . Also we give the k-prime labeling of combs Pn K1and some case of the crowns Cn K1 . Second, we show that all wheels W 2n> 1 are not k-prime for every positive integers k and n while W2n (n 2) is not k-prime for every even positive integers k . Finally, we give the k-prime labeling of the helm Hn (n 3) for 2 k 11 and we show that if k 2(n 1) and k 2n are twin primes where n 3 and k 1, then Hn is k -prime.

On the use of computer technology in the study of the discipline «Numbers theory»

Обсуждаются проблемы, связанные с компьютерными проверками теоретико-числовых гипотез. Предлагаются новые машинные способы нахождения простых чисел близнецов, совершенных и дружественных чисел, разложения Гольдбаха и проверки гипотезы Кармайкла. Problems related to computer tests of number-theoretic hypotheses are discussed. New machine methods for finding twin primes, perfect and friendly numbers, Goldbach decomposition, and testing the Carmichael hypothesis are proposed.

Solving Two Problems IN Number Theory

In 1995, Olivier Ramaret proved that any even number is the sum of no more than 6 primes. From the validity of Goldbach's ternary hypothesis (proved in 2013 year) it follows that any even number is the sum of not more than 4 numbers [1]. In the article, the author confirms the above and proves that the cause and effect of this is any even number the sum of not more than two prime and twin primes are infinite [8]-[14].

In detail explanation for the “Proof of the Twin Prime Conjecture” and its applications to thermal physics

Twin prime numbers are two prime numbers which have the difference equals to exactly 2. In other words, twin primes is a pair of two prime numbers which have the value of the difference exactly two. Sometimes the word “twin prime” is used for a pair of twin primes; an another name for this is considered as “prime twin” or called as “prime pajir”. Up to date there is no any exact proof/disproof for twin prime conjecture since roughly 200 years in the world. Through this research paper, my attempt is to provide a valid proof for twin prime conjecture. This new paper is the detailed explanation of my previous paper that I completed on mid of the year 2020 titled as ‘Proof of Twin Prime Conjecture that can be obtained by using Contradiction method in Mathematics’ (WHICH IS WELL-RECONGNIZED ALL OVER THE WORLD through researchgate as well). And this proof of the existence of infinitely many twin primes can be applied to many subject areas in Physics, Chemistry and etc. And the proof of twin prime conjecture can be used to solve several unsolved problems in Physics, Chemistry and etc as well. Also as an additional result, at the end of this research paper, it discusses about an application of the Proof of Twin Prime Conjecture to the Quantum and Thermal Physics. There, this research paper consider three space volumes symbolized as area A , B and C. Inside areas A and B there are microscopic particles separately. By applying the proof of the twin prime conjecture, finally this will try to conclude that although the areas A and B have separated by area C, there are some particles those have moved from the area B to area A (due to the high thermal pressure of area B).

In detail explanation for the “Proof of the Twin Prime Conjecture” and its applications to thermal physics

Twin prime numbers are two prime numbers which have the difference equals to exactly 2. In other words, twin primes is a pair of two prime numbers which have the value of the difference exactly two. Sometimes the word “twin prime” is used for a pair of twin primes; an another name for this is considered as “prime twin” or called as “prime pair”. Up to date there is no any exact proof/disproof for twin prime conjecture since roughly 200 years in the world. Through this research paper, my attempt is to provide a valid proof for twin prime conjecture. This new paper is the detailed explanation of my previous paper that I completed on mid of the year 2020 titled as ‘Proof of Twin Prime Conjecture that can be obtained by using Contradiction method in Mathematics’ (WHICH IS WELL-RECONGNIZED ALL OVER THE WORLD through researchgate as well). And this proof of the existence of infinitely many twin primes can be applied to many subject areas in Physics, Chemistry and etc. And the proof of twin prime conjecture can be used to solve several unsolved problems in Physics, Chemistry and etc as well. Also as an additional result, at the end of this research paper, it discusses about an application of the Proof of Twin Prime Conjecture to the Quantum and Thermal Physics. There, this research paper consider three space volumes symbolized as area A , B and C. Inside areas A and B there are microscopic particles separately. By applying the proof of the twin prime conjecture, finally this will try to conclude that although the areas A and B have separated by area C, there are some particles those have moved from the area B to area A (due to the high thermal pressure of area B).

Calculations with Constraints: Multiply and Divide by Numbers with Specific Properties

This chapter presents the techniques need to multiply or divide two numbers that have a particular property. It also introduces concepts such as twin primes and powerful numbers, and shows the classic proof that there are infinitely many primes. Applications include the size of sheets of paper, the radioactivity of bananas, the dimensions of a rugby union playing field, and the units of time and length proposed by Thomas Jefferson. The methods shown here apply mostly to numbers below, or slightly above 100. As they depend on properties of numbers, there is a need to employ pattern-spotting skills; some hints are provided.

Goldbach Conjecture, Twin Primes Conjecture, and Bounded Gap Theorem in the language of number theory

We write the formulas of the theorem and the conjectures highlighted in the title of this white paper in the language of number theory for pedagogical purpose in first-order logic.

Proof of Twin Prime Conjecture that can be obtained by using contradiction method in mathematics and it's applications to Thermal Physics

Twin prime numbers are two prime numbers which have the difference equals to 2 exactly. In other words, twin primes is a pair of two prime numbers which have the prime gap of exactly two. Sometimes the word “twin prime” is used for a pair of twin primes; an another name for this is considered as “prime twin” or called as “prime pair”. Up to date there is no any valid proof/disproof for twin prime conjecture since roughly more than 170 years in the world. Through this research paper, my attempt is to provide a valid proof for twin prime conjecture. This new paper is the detailed explanation of my previous paper that I completed on mid of the year 2020 titled as ‘Proof of Twin Prime Conjecture that can be obtained by using Contradiction method in Mathematics’ (WHICH IS WELL-RECONGNIZED ALL OVER THE WORLD through researchgate as well). And this proof of the existence of infinitely many twin primes can be applied to many subject areas in Physics, Chemistry and etc. And the proof of twin prime conjecture can be used to solve several unsolved problems in Physics, Chemistry and etc as well. Also as an additional result, at the end of this research paper, it discusses about an application of the Proof of Twin Prime Conjecture to the Quantum and Thermal Physics. There, this research paper consider three space volumes symbolized as area A , B and C. Inside areas A and B there are microscopic particles separately. By applying the proof of the twin prime conjecture, finally this will try to conclude that although the areas A and B have separated by area C, there are some particles those have moved from the area B to area A (due to the high thermal pressure of area B).