Einige S\"atze \"uber Primzahlen und spezielle binomische Ausdr\"ucke / [english] Some Theorems about prime numbers and Special Binomial expressions

Abstract

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.