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
The present paper gives the proof of the set of primes as a continuum. It starts with the density of the primes, and shortly recapitulates the prime-number-formula and the complete-prime-number-formula. Reflecting the...The present paper gives the proof of the set of primes as a continuum. It starts with the density of the primes, and shortly recapitulates the prime-number-formula and the complete-prime-number-formula. Reflecting the series of the primes over any prime gives the double density of occupation of integer positions by the union of the series of multiples of the primes. The remaining free positions render it possible to prove Goldbach’s conjecture and the set of primes as a continuum. The theoretical evaluation is followed in annexes by numerical evaluation, demonstrating the theoretical results. The numerical evaluation results in different constants and relations, which represent inherent properties of the set of primes.展开更多
The original online version of this article (Doroszlai, P. and Keller, H. (2022) The Symmetric Series of Multiples of Primes, Advances in Pure Mathematics, Vol. 12, 160-177. http://doi.org/10.4236/apm.2022.123014) unf...The original online version of this article (Doroszlai, P. and Keller, H. (2022) The Symmetric Series of Multiples of Primes, Advances in Pure Mathematics, Vol. 12, 160-177. http://doi.org/10.4236/apm.2022.123014) unfortunately contains some small mistakes. The author discovered the possible problem in Equations (1.6), (1.7) and in the last equation of the proof of lemma 2.3 and the numbering of the chapters 3 and 4 should be changed. To fix the problem, the author wishes to change these equations and titles.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
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The present paper gives the proof of the set of primes as a continuum. It starts with the density of the primes, and shortly recapitulates the prime-number-formula and the complete-prime-number-formula. Reflecting the series of the primes over any prime gives the double density of occupation of integer positions by the union of the series of multiples of the primes. The remaining free positions render it possible to prove Goldbach’s conjecture and the set of primes as a continuum. The theoretical evaluation is followed in annexes by numerical evaluation, demonstrating the theoretical results. The numerical evaluation results in different constants and relations, which represent inherent properties of the set of primes.
文摘The original online version of this article (Doroszlai, P. and Keller, H. (2022) The Symmetric Series of Multiples of Primes, Advances in Pure Mathematics, Vol. 12, 160-177. http://doi.org/10.4236/apm.2022.123014) unfortunately contains some small mistakes. The author discovered the possible problem in Equations (1.6), (1.7) and in the last equation of the proof of lemma 2.3 and the numbering of the chapters 3 and 4 should be changed. To fix the problem, the author wishes to change these equations and titles.
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.
