arXivDaily arXiv每日学术速递 周一至周五更新
arXiv周末暂无论文更新，休息一下吧，周末愉快~~
arXiv 1706.01009math.NT
收藏

关于区间中的素数

On The Prime Numbers In Intervals

Kyle D. Balliet

备注：
71 pages, Master's Thesis (2015)
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

Bertrand的假设建立了对所有正整数$n>1$，在$n$和$2n$之间存在一个素数。我们考虑这个定理的推广：对于整数$n\geq k\geq 2$，$kn$和$(k+1)n$之间是否存在素数？我们使用二项式系数和切比雪夫函数的初等方法来建立$2\leq k\leq 8$的情况。然后，我们移动到一个解析数论的方法来证明，有一个素数的间隔$(kn, (k+1)n)$至少$n\geq k$和$2\leq k\leq 519$。然后，我们考虑勒让德猜想的存在一个素数之间的$n^2$和$ (n+1)^2$的所有整数$n\geq 1$。为此，我们表明，总是有一个素数之间的$n^2$和$(n+1)^{2.000001}$的所有$n\geq 1$。此外，我们注意到，存在一个素数在区间$[n^2,(n+1)^{2+\varepsilon}]$中的任何$\varepsilon>0$和$n$足够大。我们还考虑的问题有多少素数之间的$n$和$kn$为正整数$k$和$n$为我们的每一个结果，并在一般情况下。进一步证明了$(n,kn)$中素数的个数在增加，并且对于$n\geq k\geq 2$，$(n,kn)$中至少存在$k-1$素数.最后，我们比较我们的结果，素数定理，并获得明确的下界的数量素数在我们的每一个结果。

英文摘要：

Bertrand's postulate establishes that for all positive integers $n>1$ there exists a prime number between $n$ and $2n$. We consider a generalization of this theorem as: for integers $n\geq k\geq 2$ is there a prime number between $kn$ and $(k+1)n$? We use elementary methods of binomial coefficients and the Chebyshev functions to establish the cases for $2\leq k\leq 8$. We then move to an analytic number theory approach to show that there is a prime number in the interval $(kn, (k+1)n)$ for at least $n\geq k$ and $2\leq k\leq 519$. We then consider Legendre's conjecture on the existence of a prime number between $n^2$ and $ (n+1)^2$ for all integers $n\geq 1$. To this end, we show that there is always a prime number between $n^2$ and $(n+1)^{2.000001}$ for all $n\geq 1$. Furthermore, we note that there exists a prime number in the interval $[n^2,(n+1)^{2+\varepsilon}]$ for any $\varepsilon>0$ and $n$ sufficiently large. We also consider the question of how many prime numbers there are between $n$ and $kn$ for positive integers $k$ and $n$ for each of our results and in the general case. Furthermore, we show that the number of prime numbers in the interval $(n,kn)$ is increasing and that there are at least $k-1$ prime numbers in $(n,kn)$ for $n\geq k\geq 2$. Finally, we compare our results to the prime number theorem and obtain explicit lower bounds for the number of prime numbers in each of our results.

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。

↑