Distribution of Prime Numbers

primes-utils is a Rubygem which provides a suite of utility methods to list|count primes over ranges, factoring, finding the nth prime, and primality testing.  This handbook explains the use of Prime Generators, which are used as the mathematical foundation for most of the methods, and provides the Ruby source code for the gem.

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 Hypotheses

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.

This paper develops the divisibility of the so-called Simple Primes numbers-11 (1 to 11), the discovery of a pattern to infinity, the demonstration of the Inharmonics that are 2,3,5,7 and 11 and the harmony of 1. The discovery of infinite harmony represented in fractal numbers and patterns. This is a family before the prime numbers. The simple prime numbers-11 are known as the 13-rough numbers.

This paper develops the construction of the Golden Pattern for different prime divisors, the discovery of patterns towards infinity. The discovery of infinite harmony represented in fractal numbers and patterns. These patterns form the sequence of prime numbers.

This paper, which is published in an international mathematics journal and endorsed by a mathematics society, touches on the part played by the non-trivial zeros of the Riemann zeta function ζ, providing many important information and insights in the process, including some approaches to the Riemann hypothesis.

The statements below represent the position of famous Mathematicians to this very day
"Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers and we have reason to believe that it is a mystery into which the mind will never penetrate" Leonhard Euler
"It will be another million years, at least, before we understand the Primes" Paul Erdos
Add to the above for good measure
  Terence Tao - Cryptography and Randomness of Pr...
This paper is not just for famous Mathematicians, some of which have objected to my use of Matrices and Matrix operations but also for students of Mathematics and not only

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.

Based off of Fermat's Last Theorem, a man named Andrew Beal developed a modified version of this theorem. We had to prove or disprove that for natural numbers a, b and c that have a common prime factor, the natural number exponents must be strictly greater than three.

This paper presents a simple approximation of π(x), the number of primes less than or equal to the positive real number x. Despite the high number of researchers working on it, some basic questions about primes are still open. The proposed approximation gives reliable answers to historic questions about prime numbers. It is based on the conjecture that the value of π(x)[x-π(x)]/[x^2/log x] approaches the limit 1 as x approaches infinity. When compared to the approximation formalized by the Prime Number Theorem, i.e. π(x)  ~ [x/log x], it produces results that can inspire new investigations about the nature of primes. Using the same base, approximations of the density of primes and the nth prime are also proposed.

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.

Is there an α = α end of Twin Primes = for the Twin Primes ?
α = α end of Twin Primes = a new universal constant ? !
The Twin Primes after α = α end of Twin Primes cannot disappear without being replaced by the other three categories below, as, with them, the corresponding members of [ 6α + 6 ] will disappear.
Can the Twin Primes after α = α end of Twin Primes be replaced by the other three categories above ?
Sudden boosting of the other three categories above ?
The answer is in the matrix equations below

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;

#DONATIONS. Anyone can donate for dr. Dragoi’s independent research and original music at: https://www.paypal.com/donate/?hosted_button_id=AQYGGDVDR7KH2

In this research thesis, we analyze the new possible mathematical connections between the Planck multipole spectrum data concerning the CMB, the frequencies system based on the Phi interval, various equations regarding the Theory of the Riemann zeta Function and the Theory of the distributions of primes and some sectors of String Theory


Below the link of the part II of this work:


https://www.academia.edu/44949717/On_further_possible_mathematical_connections_between_the_Planck_multipole_spectrum_data_CMB_the_frequencies_system_based_on_the_Phi_interval_various_equations_regarding_the_quantum_cosmology_the_Riemann_zeta_function_and_some_sectors_of_String_Theory_II

By means of interferences of separate prime functions, the article suggests the relation between the Riemann Z function and the distribution of prime numbers in the sense that the -1/2 real part of a nontrivial zero will be placed from the valley (or real zero point) of a cycle of an harmonic prime function different than 3 (only when it's not convergent at that point with the valley of a cycle of the function 3) to the critical strip determined by an extension of that real point in the complex plane to the imaginary point that represents the imaginary part of the non-trivial zero. When two harmonic functions interfere with a a same cycle of the function 3 (that does not have its valley at those points), no prime number will be placed there. A possible hidden asymmetry is examined there. And when there is no interference with the  amplitude of a cycle of the function 3 (there won't be any nontrivial zero) two consecutive prime numbers will be always located there. A visual geometric interpretation of the Riemann critical strip and critical line, and his trivial and non-trivial zeros is attempted to conceptually understand the Riemann Zeta function, its later generalizations,  and the Riemann hypothesis

All Primes bar 3 reside in [ 6α + 5 ] and [ 6α + 7 ] Matrices, hereinafter called Prime Residencies A specific set of Composites cohabitates with the Primes in the Prime Residencies The α coefficients of these Composites have an orderly sequence and structure Since Primes and Composites in the Primes Residencies... The α coefficients of the Primes have also an orderly sequence and structure So Euler, Erdos and current Mathematicians, like Taylor and Tao, will rejoice with the breakthrough below, if they accept that, in this case, Matrix Algebra can replace the Mathematical language they are accustomed to but very few people can understand Euler and Erdos on Primes.jpg

By means of interferences of separate prime functions, the article suggests the relation between the Riemann Z function and the distribution of prime numbers in the sense that the -1/2 real part of a nontrivial zero will be placed from the valley (or real zero point) of a cycle of an harmonic prime function different than 3 (only when it's not convergent at that point with the valley of a cycle of the function 3) to the critical strip determined by an extension of that real point in the complex plane to the imaginary point that represents the imaginary part of the non-trivial zero. When two harmonic functions interfere with a same cycle of the function 3 (that does not have its valley at those points), no prime number will be placed there. A possible hidden asymmetry that would not be follow the Riemann statement is examined there. And when there is no interference with the  amplitude of a cycle of the function 3 (there won't be any nontrivial zero) two consecutive prime numbers will be always located there. A visual geometric interpretation of the Riemann critical strip and critical line, and his trivial and non-trivial zeros is attempted to conceptually understand the Riemann Zeta function, its later generalizations,  the Riemann hypothesis and its relation with prime numbers distribution.

The paper uses Prime Generators to create progressively faster integer primality tests in Ruby. The Ruby standard library file prime.rb contains the class Integer methods prime?  and prime_division (for factorization).  I present here simpler and significantly faster methods as replacements.

(1+2) = 3, (1+3) = 4, (1+5) = 6, (1+7) = 8, (1+11) = 12, (1+13) = 14, (2+2) = 4, (2+3) = 5, (2+5) = 7, (2+7) = 9, (2+11) = 13, (2+13) = 15, (3+2) = 5, (3+3) = 6, (3+5) = 8, (3+7) = 10, (3+11) = 14, (3+13) = 16, (4+2) = 6, (4+3) = 7, (4+5) = 9, (4+7) = 11, (4+11) = 15, (4+17) = 21, Reversing order doesn't make any difference for addition. (1-2) =-1, (1-3) =-2, (1=5) =-4, (1-7) =-6, (1-11) =-10, (1-13) =-12, (2-2) = 0, (2-3) =-1, (2-5) =-3, (2-7) =-5, (2-11) =-9, (2-13) =-11, (3-2) = 1, (3-3) = 0, (3-5) =-2, (3-7) =-5, (3-11) =-9, (3-13) =-10, (4-2) = 2, (4-3) = 1, (4-5) =-1, (4-7) =-2, (4-11) =-9, (4-17) =-13,) = 1, (3-1) = 2, (7-1) = 6, (11-1) = 10, (13-1) = 12, and so on. 1,2,6,10,12, 0,1,3,5,9,11-1,0,2,5,9,10-2,-1,1,2,9,13 (1x2) = 2, (1x3) = 3, (1x5) = 5, (1x7) = 7, (1x11) = 11, (1x13) = 13, (2x2) = 4, (2x3) = 6, (2x5) = 10, (2x7) = 14, (2x11) = 22, (2x13) = 26,

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]

This paper, which is published in an international mathematics journal and endorsed by a mathematics society, touches on the part played by the non-trivial zeros of the Riemann zeta function ζ, providing many important information and insights in the process, including some approaches to the Riemann hypothesis.

In this paper, we study the symmetry between the supersingular prime number according to smallest sphenic number. With this symmetry, we show that the elements of sporadic group generates all prime numbers with the order by a simple application.

This paper presents a simple approximation of π(x), the number of primes less than or equal to the positive real number x. Despite the high number of researchers working on it, some basic questions about primes are still open. The proposed approximation gives reliable answers to historic questions about prime numbers. It is based on the conjecture that the value of π(x)[x-π(x)]/[x^2/log x] approaches the limit 1 as x approaches infinity. When compared to the approximation formalized by the Prime Number Theorem, i.e. π(x)  ~ [x/log x], it produces results that can inspire new investigations about the nature of primes. Using the same base, approximations of the density of primes and the nth prime are also proposed.

I have found a new method to instantly factorize any certain small or largest multiplied ( p*q = n) products and know its p & q factors . This method works for any small or large integers product 'n' of any digits million billion or even trillions of digits. Its all about instantly reversing (factorizing) and knowing the two factors i.e specially two small or large multiplied twin prime numbers or any terms not divisible by 2 or 3 and having gap of two.

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.