Prime Numbers Distribution Line

We prove a modification as well as an improvement of a result of K. Ford, D. R. Heath-Brown and S. Konyagin concerning prime avoidance of square-free numbers and perfect powers of prime numbers.

Prime numbers are an important research area with application in computer security. The security of certain cryptographic schemes critical in information security lies in the difficulty of factoring large numbers into their prime factors. Determination of the distribution of prime numbers is an important unsolved problem in number theory. Biomolecules such as DNA have extended the domains of what is considered a standard computer model. This report investigated the distribution of prime numbers generated from viral DNA. The digits of Euler’s number and Pi were found to be encoded between Bacteriophage T4 Watson-Crick DNA segments comprised of prime numbers. Key words: Prime number, number theory, molecular computing, phage T4.

It is practically impossible to break a cryptosystem like RSA if the key is large enough with the prime numbers are randomly distributed. Recent research shows prime numbers show a certain distribution pattern. This makes it practically possible to break such systems by solving the factoring problem. By computing the distribution probabilities and finite field incremental changes of connected prime numbers, we have validated Robert J. Lemke Oliver and Kannan Soundararajan's study findings on trailing prime numbers. We proposed and developed ranked selection algorithm on prime number distribution. We developed a number of applications, including proposing one for fast factoring the product of two large prime numbers, also known as the factoring problem.

Currently there is no known efficient formula for primes. Besides that, prime\nnumbers have great importance in e.g., information technology such as\npublic-key cryptography, and their position and possible or impossible\nfunctional generation among the natural numbers is an ancient dilemma. The\nproperties of the functions 2ab+a+b in the domain of natural numbers are\nintroduced, analyzed, and exhibited to illustrate how these single out all the\nprime numbers from the full set of odd numbers. The characterization of odd\nprimes vs. odd non-primes can be done with 2ab+a+b among the odd natural\nnumbers as an analogue to the other, well known type of fundamental\ncharacterization for irrational and rational numbers among the real numbers.\nThe prime number theorem, twin primes and erratic nature of primes, are also\ncommented upon with respect to selection, as well as with the Fermat and Euler\nnumbers as examples.\n

The Riesz means, or sometimes typical means, were introduced by M. Riesz and have been studied in connection with summability of Fourier series and of Dirichlet series [8] and [11]. In number-theoretic context, it is the Riesz sum rather than the Riesz mean that has been extensively studied. The Riesz sums appear as long as there appears the G-function. Cf. Remark 1 and [14]. As is shown below, the Riesz sum corresponds to integration while Landau's differencing is an analogue of differentiation. This integration-differentiation aspect has been the driving force of many researches on number-theoretic asymptotic formulas. Ingham's decent treatment [13] of the prime number theorem is one typical example. We state some efficient theorems that give asymptotic formulas for the sums of coefficients of the generating Dirichlet series not necessarily satisfying the functional equation.

For many centuries, people have done every possibility to find the number of prime numbers. During the 300 BC, Euclid had written a proof for there exist infinitely many prime numbers. However, the number of primes must be less than the number of integers. Hence, mathematicians started to consider precisely how many prime numbers are there comparing with all the integers. Until the end of 18th century, Gauss conjectured the total number of primes, and it was proven in the end of 19th century. The following paper will present a proof adapted from Ikehara in 1931. More precisely, the following paper converts the original problem into a better-behavior equation and then constructs another function to show the Prime Number Theorem with Function Theory and Fourier transforms. Notice that there are many more elegant proofs have been done during 20th century, and the following paper is for general understanding.

In this study, sequences are used for direct computation of prime numbers. A single equation for generating all prime numbers with the exception of 2 and 3 and composite numbers that are not divisible by 2 or 3 is presented. A periodicity to the indices of the composites generated by that equation is determined. An equation to determine the indices of the composite numbers is derived. The equation for determining the composite indices is then altered to avoid redundancy by observation of its diagonal, when its values are inserted into a matrix that has the indexes (j) for columns and (k) for rows. The two equations are presented in several different forms and are used in conjunction to render a technique for computing prime numbers indefinitely. The validity of the technique is proven up to one billion by computation and the prime number theorem.

In this paper, we analyze properties of prime number sequences produced by the alternating sum of higher-order subsequences of the primes. We also introduce a new sieve which will generate these prime number sequences via the systematic selection and elimination of prime number indices on the real number line.

On the difference between consecutive prime numbers

On the order of magnitude of the difference between consecutive prime numbers

We give a quick tour through some topics in analytic prime number theory, focusing in particular on the strange mixture of order and chaos in the primes. For instance, while primes do obey some obvious patterns (e.g. they are almost all odd), and have a very regular asymptotic distribution (the prime number theorem), we still do not know a deterministic formula to quickly generate large numbers guaranteed to be prime, or to count even very simple patterns in the primes, such as twin primes p,p+2. Nevertheless, it is still possible in some cases to understand enough of the structure and randomness of the primes to obtain some quite nontrivial results.

The study of the primes is one of the most important branches of number theory. The only other branch of similar importance is Diophantine equations and this will be considered in the final three chapters. The literature on the prime numbers is very extensive beginning with Pythagoras and Euclid; even so, many problems remain. There are two pre-eminent results: Dirichlet’s theorem and the prime number theorem (PNT).

We discuss the formalization, in the Matita Interactive Theorem Prover, of some results by Chebyshev concerning the distribution of prime numbers, subsuming, as a corollary, Bertrand's postulate. Even if Chebyshev's result has been later superseded by the stronger prime number theorem, his machinery, and in particular the two functions psi and theta still play a central role in the modern development of number theory. The proof makes use of most part of the machinery of elementary arithmetics, and in particular of properties of prime numbers, gcd, products and summations, providing a natural benchmark for assessing the actual development of the arithmetical knowledge base.

In this paper it is proposed and proved an exact formula for the prime-counting function, finding an expression of Legendre's formula. As corollaries, they are proved some important conjectures regarding prime numbers distribution.

The Bateman–Horn conjecture: Heuristic, history, and applications