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

TL;DR

This paper presents new theorems on prime numbers, including their distribution in quadratic intervals, laws governing twin primes and Goldbach pairs, and provides proofs for the prime number theorem.

Contribution

It introduces novel theorems about prime distribution, twin primes, and Goldbach pairs, along with new proofs for the prime number theorem.

Findings

No quadratic interval contains fewer than 2 primes.

The number of primes up to n follows an exact law approximating Gauss's prime number theorem.

The number of twin primes and Goldbach pairs tends to infinity as n increases.

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.