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

Abstract

Let $n\in {\mathbb{Z}}^{+}$. Is it true that every sequence of $n$ consecutive integers greater than $n^2$ and smaller than $(n+1{)}^2$ contains at least one prime number? In this paper 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 $[n,n+2\lfloor \sqrt{n}\rfloor -1]$.

11 pages; the title and the abstract have been modified; a few references have been added, as it has been suggested to the author

References in corpus (2)

• Estimates of Some Functions Over Primes without R.H.
• On the Primes in the Interval [3n, 4n]

Cited by in corpus (1)

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