1. There is no existing any quadratic interval $\eta_{ n}:=(n^{ 2},(n+1)^{ 2}],$ which contains less than 2 prime numbers. The number of prime numbers within $\eta_{ n}$ goes averagely linear with n to infinity. 2. The exact law of the number $\pi(n)$ of prime numbers smaller or equal to n is given. As an approximation of that we get the prime number theorem of Gauss for great values of n. 3. We derive partition laws for $\pi(\eta_{ n})$, for the number of twin primes $\pi_{ 2}(\eta_{ n})$ in quadratic intervals $\eta_{ n}$ and for the multiplicity $\pi_{ g}(2n)$ of representations of Goldbach-pairs for a given even number 2n similiar to the theorem of Gauss. 4. There is no natural number n>7, which is beginning point of a prime number free interval with a length of more than 2{ *}SQRT(n). 5. It follows, that the number of twin primes goes to infinity as well as the number of Goldbach-pairs for a given 2n, if n goes to infinity. 6. Besides this our computation gives new proofs for the prime number theorem of Gauss.
Categories: math.GM