Einige Sätze über Primzahlen und spezielle binomische Ausdrücke / [english]关于素数和特殊二项式的几个定理
Einige Sätze über Primzahlen und spezielle binomische Ausdrücke / [english] Some Theorems about prime numbers and Special Binomial expressions
摘要：
1.不存在含有少于2个素数的二次区间$η_{n}:=(n^{2},(n+1)^{2}],$。在$η_{n}$中的素数的数量与n成平均线性关系，直到无穷大。2.给出了小于或等于n的素数的个数$π(n)$的精确规律。作为这个定理的一个近似，我们得到了n为大值时高斯的素数定理。3.本文导出了$π(η_{n})$的配分律，二次区间$η_{n}$中孪生素数个数$π_{2}(η_{n})$的配分律，以及哥德巴赫对表示的重数$π_{g}(2n)$的配分律。4.不存在自然数n>7，这是长度大于2{*}SQRT（n）的无素数区间的起点。5.因此，孪生素数的数量会趋于无穷大，对于给定的2n，如果n趋于无穷大，哥德巴赫对的数量也会趋于无穷大。6.此外，我们的计算还对高斯的素数定理给出了新的证明.
英文摘要：
1. There is no existing any quadratic interval $η_{n}:=(n^{2},(n+1)^{2}],$ which contains less than 2 prime numbers. The number of prime numbers within $η_{n}$ 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 $π(η_{n})$, for the number of twin primes $π_{2}(η_{n})$ in quadratic intervals $η_{n}$ and for the multiplicity $π_{g}(2n)$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
