Distribution of Prime Numbers

The fundamental theorem of arithmetic states that any composite natural integer can be expressed in one and only one way as a product of prime numbers. This sets the understanding of the organization of prime numbers at the core of number theory. In this work we present a simple, self-consistent and deterministic scheme allowing to investigate the intrinsic organization of prime numbers. Using this scheme, we establish an algorithm to compute-without any primality test-the complete list of prime numbers below any preassigned limit x. Counting the latter yields π(x), the number of prime numbers below x. Based on preliminary tests on computing clusters available, a considerable gain in computational speed and algorithmic simplicity towards producing complete lists of large prime numbers is observed. At the core of the new scheme lays its ability to provide, in a deterministic way, complete lists of consecutive and composite odd numbers below any preassigned limit x. The complete list of prime numbers below x is deduced from the latter. The two key ingredients of the scheme are a set of eleven generic tables, coupled with a three-criteria test applied on the differences between pairs of the consecutive composite odd numbers initially obtained. Being able to count all the elements of the deduced list of prime numbers means that our deterministic scheme provides a new approach to the long standing problem of " how many prime numbers are there below any preassigned limit x " : one of Gauss's Dreams. Our deterministic scheme therefore contributes towards studies aimed at unveiling the organization of prime numbers. We illustrate the latter in a follow-up paper, Paper II [7], where we offer a new perspective on the Riemann hypothesis. NOTA BENE: The present copyrighted work is made public for constructive comments and NOT for mass dissemination. Please feel free to contact the authors with constructive comments.

When considering whether every even integer can be expressed as the sum of two primes, it is tempting to view the puzzle as a question of arithmetic, while the answer lies in the infinite pattern of the primes. Instead of attempting to prove that every even integer has this property, and ignoring the noticeable pattern that larger even numbers have more prime pairs, we attempt to find an even number without this property, consider how many of these such numbers can exist, and come to the conclusion that no such number can exist. Because it is simple to show that small numbers (no more than three digits) have the property, we are most concerned with larger numbers, considering whether the widening gaps between larger primes may eventually create an even number that cannot be expressed between the sum of the plentiful small primes and the highly scarce large primes. In our exploration, we will mathematically create the ideal scenario for a non-prime-sum number to exist, then show that as this scenario extends towards the infinite, a non-prime-sum number does not become more possible, but in fact less possible. We will additionally see that extending toward the infinite, the number of prime pairs that sum to a given even number will increase without limit.

Let p(n) be the nth prime and p(p(n)) be the nth prime-indexed prime (PIP). The process of taking prime-indexed subsequences of primes can be iterated, and the number of such iterations is the prime-index order. We report empirical evidence that the set composed of finite-differenced PIP sequences of prime-index order k >= 1 forms a quasi-self-similar fractal structure with scaling by prime-index order. Strong positive linear correlation (r >= 0.926) is observed for all pairwise combinations of these finite-differenced PIP sequences over the range of our sample, the first 1.3 billion primes. The structure exhibits translation invariance for shifts in the index set of the PIP sequences. Other free parameters of the structure include prime-index order and the order and spacing of the finite difference operator. The structure is graphed using 8-bit color fractal plots, scaled across prime-index orders k = 1..6 and spans the first 1.3 billion primes.

El descubrimiento de un patrón para números primos menores que 11. Este Patrón presenta características sorprendentes, funciona en dos columnas opuestas complementarias en las cuales se reparten las cantidades en forma simétrica y en total armonía. El Patrón Dorado tiene características sorprendentes.

This paper describes the mathematical foundation of Prime Generators and their use in creating a fast Twin Primes Segmented Sieve of Zakiya (SSoZ), and also their applications to Number Theory, including Mersenne Primes, creating an exact Prime-counting Function, and implications for the Riemann Hypothesis.

Constantine Adraktas has proven, on the basis of the fundamental theorem of arithmetic, that the primes are not random. Additionally he has discovered Matrix equations for them.
This is the ultimate proof that the Prime Numbers are not Random Numbers as famous Mathematicians believe and claim publicly through presentations you can find on U Tube
Any pupil around the world and a wide public will understand the first 2 pages of the paper below.  Feel free to ask questions by emailing me [email protected]

Let p n be the nth prime and p p n be the nth prime-indexed prime (PIP). The process of taking prime-indexed subsequences of primes can be iterated, and the number of such iterations is the prime-index order. We report empirical evidence that the set composed of finite-differenced PIP sequences of prime-index order k ≥ 1 forms a quasiself-similar fractal structure with scaling by prime-index order. Strong positive linear correlation (r ≥ 0.926) is observed for all pairwise combinations of these finite-differenced PIP sequences over the range of our sample, the first 1.3 billion primes. The structure exhibits translation invariance for shifts in the index set of the PIP sequences. Other free parameters of the structure include prime-index order and the order and spacing of the finite difference operator. The structure is graphed using 8-bit color fractal plots, scaled across prime-index orders k = 1..6 and spans the first 1.3 billion primes.

Euler’s formula establishes the relationship between the trigonometric function and the exponential function. In doing so unifies two waves, a real and an imaginary one, that propagate through the Complex number set, establishing relation between integer numbers. A complex wave, if anchored by zero and by a defined integer number N, only can assume certain oscillation modes. The first mode of oscillation corresponds always to a N prime number and the other modes to its multiples.
ψ(x) = xei(nπ N x)
Under the above described conditions, these waves and their admissible oscillation modes allows for primality testing of integer numbers, the deduction of a new formula π(x) for counting prime numbers and the identification of patterns in the prime numbers distribution with computing time gains in the calculations. In this article, four theorems and one rule of factorizing are put forward with consequences for prime number signaling, counting and distribution. Furthermore, it is establish the relationship between this complex wave with a time independent semi-classical harmonic oscillator, in which the spectrum of the allowed energy levels are always only prime numbers. Thus, it is affirmative the reply to the question if the prime numbers distribution is related to the energy levels of a physical system.

This work proposes the generalization of the binary (strong) Goldbach’s Conjecture (BGC), briefly called “the Vertical Binary Goldbach’s Conjecture” (VBGC), which is essentially a meta-conjecture, as VBGC states an infinite number of conjectures stronger than BGC, which all apply on “iterative” primes with recursive prime indexes (named “i-primeths” in this article, as derived from the concept of generalized “primeths”, a term first introduced by N. J. A. Sloane and Robert G. Wilson in their “primeth recurrence” concept). VBGC was discovered by the author of this paper in 2007 and perfected (by computational verifications) until 2017 by using the arrays of matrices of Goldbach index-partitions (GIPs), which are a useful tool in studying BGC by focusing on prime indexes. This work also brings in a synthesized classification of some Goldbach-like conjectures (GLC) (including those which are “stronger” than BGC) and a new class of GLCs “stronger” than BGC, from which VBGC distinguishes as a very important conjecture of primes (with potential importance in the optimization of the BGC experimental verification and other possible theoretical and practical applications in mathematics and physics) and a very special self-similar property of the primes subset of positive integers. Keywords: Prime (number), primes with prime indexes aka prime-index primes (PIPs), i-primeths (i-PIPs), the Binary Goldbach Conjecture (BGC), Goldbach index-partition (GIP), Goldbach-like conjectures (GLC), the Vertical Binary Goldbach Conjecture (VBGC) as applied on i-primeths;

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