The present paper gives the proof of the set of primes as continuum and evaluates the analytical formula for the integral of the inverse of the primes over the distance. First it starts with the density of the primes,...The present paper gives the proof of the set of primes as continuum and evaluates the analytical formula for the integral of the inverse of the primes over the distance. First it starts with the density of the primes, shortly recapitulates the prime-number-formula and the complete-prime-number-formula, the proof of the set of primes as continuum. The theoretical evaluation is followed in annexes by numerical evaluation of the theoretical results and of different constants, which represent inherent properties of the set of primes.展开更多
In this paper we give an algorithm for polignac numbers. That algorithm is also the base of the proof of the de Polignac's conjecture. The examples of the application present algorithm for twin, cousin, and sexy p...In this paper we give an algorithm for polignac numbers. That algorithm is also the base of the proof of the de Polignac's conjecture. The examples of the application present algorithm for twin, cousin, and sexy primes are included.展开更多
It is generally known that under the generalized Riemann hypothesis one could establish the twin primes conjecture by the circle method, provided one could obtain the estimate o (nlog-2 n)?for the integral of the repr...It is generally known that under the generalized Riemann hypothesis one could establish the twin primes conjecture by the circle method, provided one could obtain the estimate o (nlog-2 n)?for the integral of the representation function over the minor arcs. One of the new results here is that the assumption of GRH can be removed. We compare this and other such sufficiency results with similar results for the Goldbach conjecture.展开更多
The union of the straight and over the point of reflection—reflected series of the arithmetic progression of primes results in the double density of occupation of integer positions. It is shown that the number of fre...The union of the straight and over the point of reflection—reflected series of the arithmetic progression of primes results in the double density of occupation of integer positions. It is shown that the number of free positions left by the double density of occupation has a lower limit function, which is growing to infinity. The free positions represent equidistant primes to the point of reflection: in case the point of reflection is an even number, they satisfy Goldbach’s conjecture. The double density allows proving as well that at any distance from the origin large enough—the distance between primes is smaller, than the square root of the distance to the origin. Therefore, the series of primes represent a continuum and may be integrated. Furthermore, it allows proving that the largest gap between primes is growing to infinity with the distance and that the number of any two primes, with a given even number as the distance between them, is unlimited. Thus, the number of twin primes is unlimited as well.展开更多
The union of the straight and—of the over a point of reflection—reflected union of the series of the arithmetic progression of primes results the double density of occupation of integer positions by multiples of the...The union of the straight and—of the over a point of reflection—reflected union of the series of the arithmetic progression of primes results the double density of occupation of integer positions by multiples of the primes. The remaining free positions represent diads of equidistant primes to the point of reflection: in case the point of reflection is an even number, they satisfy Goldbach’s conjecture. Further, it allows to prove, that the number of twin primes is unlimited. The number of all greater gaps as two between primes has well defined lower limit functions as well: it is evaluated with the local density of diads, multiplied with the total of the density of no-primes of all positions over the distance between the components of the diads (the size of the gaps). The infinity of these lower limit functions proves the infinity of the number of gaps of any size between primes. The connection of the infinite number of diads to the infinity of the number of gaps of any size is the aim of the paper.展开更多
A simple recursive algorithm to generate the set of natural numbers, based on Mersenne numbers: M<sub>N</sub> = 2<sup>N</sup> – 1, is used to count the number of prime numbers within the preci...A simple recursive algorithm to generate the set of natural numbers, based on Mersenne numbers: M<sub>N</sub> = 2<sup>N</sup> – 1, is used to count the number of prime numbers within the precise Mersenne natural number intervals: [0;M<sub>N</sub>]. This permits the formulation of an extended twin prime conjecture. Moreover, it is found that the prime numbers subsets contained in Mersenne intervals have cardinalities strongly correlated with the corresponding Mersenne numbers.展开更多
文摘The present paper gives the proof of the set of primes as continuum and evaluates the analytical formula for the integral of the inverse of the primes over the distance. First it starts with the density of the primes, shortly recapitulates the prime-number-formula and the complete-prime-number-formula, the proof of the set of primes as continuum. The theoretical evaluation is followed in annexes by numerical evaluation of the theoretical results and of different constants, which represent inherent properties of the set of primes.
文摘In this paper we give an algorithm for polignac numbers. That algorithm is also the base of the proof of the de Polignac's conjecture. The examples of the application present algorithm for twin, cousin, and sexy primes are included.
文摘It is generally known that under the generalized Riemann hypothesis one could establish the twin primes conjecture by the circle method, provided one could obtain the estimate o (nlog-2 n)?for the integral of the representation function over the minor arcs. One of the new results here is that the assumption of GRH can be removed. We compare this and other such sufficiency results with similar results for the Goldbach conjecture.
文摘The union of the straight and over the point of reflection—reflected series of the arithmetic progression of primes results in the double density of occupation of integer positions. It is shown that the number of free positions left by the double density of occupation has a lower limit function, which is growing to infinity. The free positions represent equidistant primes to the point of reflection: in case the point of reflection is an even number, they satisfy Goldbach’s conjecture. The double density allows proving as well that at any distance from the origin large enough—the distance between primes is smaller, than the square root of the distance to the origin. Therefore, the series of primes represent a continuum and may be integrated. Furthermore, it allows proving that the largest gap between primes is growing to infinity with the distance and that the number of any two primes, with a given even number as the distance between them, is unlimited. Thus, the number of twin primes is unlimited as well.
文摘The union of the straight and—of the over a point of reflection—reflected union of the series of the arithmetic progression of primes results the double density of occupation of integer positions by multiples of the primes. The remaining free positions represent diads of equidistant primes to the point of reflection: in case the point of reflection is an even number, they satisfy Goldbach’s conjecture. Further, it allows to prove, that the number of twin primes is unlimited. The number of all greater gaps as two between primes has well defined lower limit functions as well: it is evaluated with the local density of diads, multiplied with the total of the density of no-primes of all positions over the distance between the components of the diads (the size of the gaps). The infinity of these lower limit functions proves the infinity of the number of gaps of any size between primes. The connection of the infinite number of diads to the infinity of the number of gaps of any size is the aim of the paper.
文摘A simple recursive algorithm to generate the set of natural numbers, based on Mersenne numbers: M<sub>N</sub> = 2<sup>N</sup> – 1, is used to count the number of prime numbers within the precise Mersenne natural number intervals: [0;M<sub>N</sub>]. This permits the formulation of an extended twin prime conjecture. Moreover, it is found that the prime numbers subsets contained in Mersenne intervals have cardinalities strongly correlated with the corresponding Mersenne numbers.
