On primes amongst squares of naturals

TL;DR

This paper investigates the distribution of primes related to squares of natural numbers, examining inequalities involving square roots, prime occurrence in intervals, and primes between consecutive squares.

Contribution

It introduces new insights into prime distribution around squares of natural numbers and explores inequalities and intervals concerning primes.

Findings

The inequality rac{P_{k+1}}{P_k}<1 is analyzed.

Existence of primes in arbitrary real intervals is discussed.

Primes between consecutive squares are addressed.

Abstract

We analyze the inequality $\sqrt{P_{k+1}}-\sqrt{P_{k}}<1,\ k\in\mathbb{N}$, discuss the existence of primes on arbitrary intervals $(r,s),\ r<s,\ r,s\in\mathbb{R}$, and finally address the issue of primes between squares of naturals.