Einige Sätze über Primzahlen und spezielle binomische Ausdrücke / [english] Some Theorems about prime numbers and Special Binomial expressions

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1. There is no existing any quadratic interval ηₙ:=(n²,(n+1)²], which contains less than 2 prime numbers. The number of prime numbers within ηₙ goes averagely linear with n to infinity. 2. The exact law of the number π(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 π(ηₙ), for the number of twin primes π₂(ηₙ) in quadratic intervals ηₙ and for the multiplicity πg(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.