The Gaps between Primes

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.

The Number of Primes

It is known that the prime-number-formula at any distance from the origin has a systematic error. It is shown that this error is proportional to the square of the number of primes present up to the square root of the distance. The proposed completion of the prime-number-formula in the present paper eliminates this systematic error. This is achieved by using a quickly converging recursive formula. The remaining error is reduced to a symmetric dispersion of the effective number of primes around the completed prime-number-formula. The standard deviation of the symmetric dispersion at any distance is proportional to the number of primes present up to the square root of the distance. Therefore, the absolute value of the dispersion, relative to the number of primes is approaching zero and the number of primes resulting from the prime-number-formula represents the low limit of the number of primes at any distance.

The Symmetric Series of Multiples of Primes

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.