Twin Prime Numbers Research Articles

Analyzing twin primes, Goldbach's strong conjecture and Polignac's conjecture

Here we analyze three well-known conjectures: (i) the existence of infinitely many twin primes, (ii) Goldbach's strong conjecture, and (iii) Polignac's conjecture. We show that the three conjectures are related to each other. In particular, we see that in analysing the validity of Goldbach's strong conjecture, one must consider also the existence of an infinite number of twin primes. As a consequence of how we approach this analysis, we also observe that if this conjecture is true, then so is Polignac's conjecture. Our first step is an analysis of the existence of infinitely many twin prime numbers. For this, using the formula 4((n−1)!+1) ≡−n (mod n(n + 2)) – satisfied if and only if (n, n + 2) are twin primes –, together with Wilson's theorem, we obtain conditions that must be met for two numbers to be twin primes. Our results, obtained from an analytic and functional study, lead us to conclude that there may exist infinitely many twin primes. Next, we consider the validity of Goldbach's strong conjecture. After showing that the conjecture is true for the first even numbers, we notice a pattern that we analyze for any even number, reducing it to three cases: (i) when the even number 2n is two times a prime number n; (ii) when the even number 2n is such that n=2m, with 2m-1 and 2m+1 twin primes; (iii) all other cases, i.e., for any 2n even number ∀n∈N with n > 1, n prime or not, with independence of n=2m being 2m-1 and 2m+1 twin primes or not. In this last case, we show that one can always find a certain r∈N such that 1 < r < n satisfying that n − r and n + r are primes, so that their sum is 2n. In this case, we use the reduction to absurd method, and our results lead us to conclude that Goldbach's strong conjecture is true to the best of our calculations, and Polignac's conjecture as well.

Preliminary Identification of a Prime Number Other Than 2 and 3, the Origin of Twin Prime Numbers, the Structure of the Chain of Prime Numbers and the Set of Prime Numbers Less Than a Given Integer

Preliminary Identification of a Prime Number Other Than 2 and 3, the Origin of Twin Prime Numbers, the Structure of the Chain of Prime Numbers and the Set of Prime Numbers Less Than a Given Integer

THE 5 EQUATIONS THAT GENERATE ALL COMPOSITE NUMBERS

This paper presents 5 equations that generate all existing composite numbers. To find these equations, we divided all natural numbers in 6 groups and found the equations that generate all composite numbers in each one of these groups by means of simple mathematical reasoning. Two different ways to obtain prime numbers by iteration are provided. An iteration method to find twin prime numbers is also described.

Special Spirals are Produced by the ROTASE Galactic Spiral Equations with the Sequential Prime Numbers

In this paper, the sequential prime numbers are used as variables for the galactic spiral equations which were developed from the ROTASE model. Special spiral patterns are produced when prime numbers are treated with the unit of radian. The special spiral patterns produced with the first 1000 prime numbers have 20 spirals arranged in two groups. The two groups have perfect central symmetry with each other and are separated with two spiral gaps. The special spiral pattern produced with natural numbers from 1 to 7919 shows 6 spirals in the central area and 44 spirals in the outer area. The whole 7919 spiral points can be plotted with either 6-spiral pattern or 44-spiral pattern. The special spiral pattern is well explained with careful analysis, it is concluded that all prime numbers greater than 3 must meet one of the equations: P1 = 1 + 6 * n (n > 0, n is an integer) P5 = 5 + 6 * m (m ≥ 0, m is an integer) Matching one of the equations is a necessary condition for a number to be a prime number, not a sufficient condition. Twin prime numbers can only be formed between P1 and P5 prime numbers, n must be 1 greater than m. The largest prime number is known at the moment 2^(82,589,933) – 1 is a P1 prime number.

Some Results on Number Theory and Analysis

In this work, we obtain bounds for the sum of the integer solutions of quadratic polynomials of two variables of the form <img src=image/13423078_01.gif> where <img src=image/13423078_02.gif> is a given natural number that ends in one. This allows us to decide the primality of a natural number <img src=image/13423078_02.gif> that ends in one. Also we get some results on twin prime numbers. In addition, we use special linear functionals defined on a real Hilbert space of dimension <img src=image/13423078_03.gif> , in which the relation is obtained: <img src=image/13423078_04.gif>, where <img src=image/13423078_05.gif> is a real number for <img src=image/13423078_06.gif>. When <img src=image/13423078_07.gif> or <img src=image/13423078_08.gif>, we manage to address Fermat's Last Theorem and the equation <img src=image/13423078_09.gif>, proving that both equations do not have positive integer solutions. For <img src=image/13423078_10.gif>, the Cauchy-Schwartz Theorem and Young's inequality were proved in an original way.

Twin Prime Distribution Problem

The distribution of twin prime numbers is discussed. The research method of corresponding prime number distribution is proposed. The distribution of prime numbers corresponding to integers and composite numbers is discussed. Through the corresponding prime distribution rate of integers and composite numbers, it is found that the corresponding prime distribution rate of composite numbers approaches the corresponding prime distribution rate of integers. The distribution principle of corresponding prime number of composite number is proved. The twin prime distribution theorem is obtained. The number of twin prime numbers is thus obtained. It provides a practical way to study the conjecture of twin prime numbers.

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 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 proof for twin prime conjecture.

A Solution to the Famous “Twin’s Problem”

In the following pages I will try to give a solution to this very known unsolved problem of theory of numbers. The solution is given here with an important analysis of the proof of formula (4.18), with the introduction of special intervals between square of prime numbers that I call silver intervals . And I make introduction of another also new mathematic phenomenon of logical proposition “In mathematics nothing happens without reason” for which I use the ancient Greek term “catholic information”. From the theorem of prime numbers we know that the expected multitude of prime numbers in an interval is given by formula considering that interval as a continuous distribution of real numbers that represents an elementary natural numbers interval. From that we find that in the elementary interval around of a natural number ν we easily get by dx=1 the probability that has the ν to be a prime number. From the last formula one can see that the second part of formula (4.18) is absolutely in agreement with the above theorem of prime numbers. But the benefit of the (4.18) is that this formula enables correct calculations in set N on finding the multitude of twin prime numbers, in contrary of the above logarithmic relation which is an approximation and must tend to be correct as ν tends to infinity. Using the relationship (4.18) we calculate here the multitude of twins in N, concluding that this multitude tends to infinite. But for the validity of the computation, the distribution of the primes in a random silver interval is examined, proving on the basis of catholic information that the density of primes in the same random silver interval is statistically constant. Below, in introduction, we will define this concept of “catholic information” stems of “information theory” [1] and it is defined to use only general forms in set N, because these represent the set N and not finite parts of it. This concept must be correlated to Riemann Hypothesis.

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.

Twin Prime Numbers and Diophantine Equations

For two millennia, the prime numbers have continued to fascinate mathematicians. Indeed, a conjecture which dates back to this period states that the number of twin primes is infinite. In 1949 Clement showed a theorem on twin primes. Starting from Wilson's theorem, Clement’s theorem and the corollary of Clement’s theorem [1], I came to find Diophantine equations whose solution could lead to theproof of the infinitude of twin primes.

Construction of thinned phased antenna arrays by coding the positions of array elements in a equidistant grid of nodes on the basis of the properties of prime and composite number moduli

A method for construction of thinned phased antenna arrays (PAAs) by coding the positions of array elements in an equidistant grid of nodes on the basis of the properties of prime and composite number moduli is proposed. (Such arrays are abbreviated as PAA-C.) An algorithm for the arrangement of array elements is described, examples of the arrangement of elements are presented, and the array patterns and their cross sections are calculated. A method for suppression of sidelobes of the PAA-C pattern is proposed. The method is based on the multiplication of the patterns of PAA-Cs differing in their codes corresponding to one modulus or in coding by twin prime numbers.

Programming, Tuning and Automatic Parallelization of Irregular Divide-and-Conquer Applications in DAMPVM/DAC

This paper presents a new object oriented framework DAMPVM/DAC which is implemented on top of DAMPVM and provides automatic partitioning of irregular divide-and-conquer (DAC) applications at runtime. The processes are then mapped dynamically to processors taking into account their speeds and even loads by other user processes. The paper presents the programming interface (API) of the framework, available tuning mechanisms, internal solutions with respect to automatic partitioning and mapping to processors. Finally, specific parameters, optimization techniques and simulation results are shown for a variety of irregular divide-and-conquer applications. The applications include αβ search, recursive Fibonacci, ([UNKNOWN]), finding twin prime numbers in parallel and previously implemented and now carefully analyzed and tuned adaptive quadrature integration and image recognition.Various DAC parameters were tuned for specific applications including costs of computing vectors/subtrees, maximum partitioning levels etc. Moreover, the overhead of DAMPVM/DAC compared to sequential implementations is shown for all the implemented applications.

A Relation between Prime Numbers and Twin Prime Numbers

Every mathematician has been concerned with prime numbers, and has metwith mysterious surprises about them. Besides intuition, using empirical methods has an important role to findrelations between prime numbers. A relation between any prime numberand any twin prime number has been obtained.

A class of primality criteria formulated in terms of the divisibility of binomial coefficients

We find a class of theorems of the type “q is a prime number iff R(q) is a divisor of the binomial coefficient\(\left( {\begin{array}{*{20}c} {S(q)} \\ {T(q)} \\ \end{array} } \right)\) ”; here R, S, T are certain fully significant functions that are superpositions of addition, subtraction, multiplication, division, and raising to a power. Similar criteria were also obtained for prime Mersenne numbers, prime Fermat numbers, and twin-prime numbers.

An Investigation of the Average Distribution of Twin Prime Numbers

An Investigation of the Average Distribution of Twin Prime Numbers