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).

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).

Proof of Twin Prime Conjecture that can be obtained by using Contradiction method in Mathematics

Twin prime numbers are two prime numbers which have the difference of 2 exactly. In other words, twin primes is a pair of prime that has a prime gap of two. Sometimes the part “twin prime” is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Up to date there is no any valid proof/disproof for twin prime conjecture. Through this research paper, my attempt is to provide a valid proof for twin prime conjecture.

Disproof of Twin Prime Conjecture

A twin prime numbers are two prime numbers which have the difference of 2 exactly. In other words, twin primes is a pair of prime that has a prime gap of two. Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Up to date there is no any valid proof/disproof for twin prime conjecture. Through this research paper, my attempt is to provide a valid disproof for twin prime conjecture.

Additively unique sets of prime numbers

Spiro proved that the identity function is the only multiplicative function with [Formula: see text] for some prime [Formula: see text] and [Formula: see text] for all prime [Formula: see text] and [Formula: see text]. We determine the sets [Formula: see text] of primes for which restricting our condition to [Formula: see text] for all [Formula: see text] still implies that [Formula: see text] is the identity function. We prove that [Formula: see text] satisfies these conditions if and only if [Formula: see text] contains every prime that is not the larger element of a twin prime pair and [Formula: see text] contains [Formula: see text] or [Formula: see text].

R-prime numbers of degree k

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.

Markov partitions and homology for -solenoids

Given a relatively prime pair of integers, $n\geq m>1$, there is associated a topological dynamical system which we refer to as an $n/m$-solenoid. It is also a Smale space, as defined by David Ruelle, meaning that it has local coordinates of contracting and expanding directions. In this case, these are locally products of the real and various $p$-adic numbers. In the special case, $m=2,n=3$ and for $n>3m$, we construct Markov partitions for such systems. The second author has developed a homology theory for Smale spaces and we compute this in these examples, using the given Markov partitions, for all values of $n\geq m>1$ and relatively prime.