Two statements that are equivalent to a conjecture related to the distribution of prime numbers

Abstract

Let $n\in {\mathbb{Z}}^{+}$. In [8] we ask the question whether any sequence of $n$ consecutive integers greater than $n^2$ and smaller than $(n+1{)}^2$ contains at least one prime number, and we show that this is actually the case for every $n\le 1,193,806,023$. In addition, we prove that a positive answer to the previous question for all $n$ would imply Legendre's, Brocard's, Andrica's, and Oppermann's conjectures, as well as the assumption that for every $n$ there is always a prime number in the interval $[n,n+2\lfloor \sqrt{n}\rfloor -1]$. Let $\pi [n+g(n),n+f(n)+g(n)]$ denote the amount of prime numbers in the interval $[n+g(n),n+f(n)+g(n)]$. Here we show that the conjecture described in [8] is equivalent to the statement that $$\pi [n+g(n),n+f(n)+g(n)]\ge 1\text{, }\forall n\in {\mathbb{Z}}^{+}\text{,}$$ where $$f(n)=\left(\frac{n-\lfloor \sqrt{n}{\rfloor }^2-\lfloor \sqrt{n}\rfloor -\beta }{|n-\lfloor \sqrt{n}{\rfloor }^2-\lfloor \sqrt{n}\rfloor -\beta |}\right)(1-\lfloor \sqrt{n}\rfloor )\text{, }g(n)=\lfloor 1-\sqrt{n}+\lfloor \sqrt{n}\rfloor \rfloor \text{,}$$ and $\beta$ is any real number such that $1<\beta <2$. We also prove that the conjecture in question is equivalent to the statement that $$\pi [S_n,S_n+\lfloor \sqrt{S_n}\rfloor -1]\ge 1\text{, }\forall n\in {\mathbb{Z}}^{+}\text{,}$$ where $$S_n=n+\frac{1}{2}{\lfloor \frac{\sqrt{8n+1}-1}{2}\rfloor }^2-\frac{1}{2}\lfloor \frac{\sqrt{8n+1}-1}{2}\rfloor +1\text{.}$$ We use this last result in order to create plots of $h(n)=\pi [S_n,S_n+\lfloor \sqrt{S_n}\rfloor -1]$ for many values of $n$.

16 pages, 3 figures (version 2 includes them also as ancillary files, no changes to version 1 have been made), Mathematica code; keywords: Andrica's conjecture, Brocard's conjecture, Legendre's conjecture, Oppermann's conjecture, prime numbers, triangular numbers. arXiv admin note: text overlap with arXiv:1310.1323

References in corpus (1)

• On Legendre's, Brocard's, Andrica's, and Oppermann's Conjectures