The Infinity of the Number of Pairs of Equidistant Primes around Any Integer over Three

Abstract

From Goldbach’s conjecture to Goldbach’s law, the present proof is based on reflecting the series of the primes over any integer, giving the double density of occupation of integer positions. The remaining free positions are diads, equidistant primes to the point of reflection. Based on the symmetry due to the reflection different proofs are given on Goldbach’s conjecture. One of the proofs requires the asymptotically exact number of primes up to a growing number of integers. Based on the prime-number-formula a correction as the best estimating function of the number of primes, the complete-prime-number-formula is evaluated. Similarly to the primes, a diads-number formula is evaluated. The dispersion of the effective number of primes and diads around the corresponding best estimation functions allows us to prove the asymptotic continuity of both functions. The asymptotic continuity of the best estimate functions the prime-number-formula and the diads-number formula may be proved as low limit functions of the effective number of primes and diads. This gives the first proof of Goldbach’s conjecture. Two others follow. 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.