On the Firoozbakht's conjecture

Abstract

This paper proves Firoozbakht's conjecture using Rosser and Schoenfelds' inequality on the distribution of primes. This inequality is valid for all natural numbers ${n\geq 21}$. Firoozbakht's conjecture states that if $ {p_{n}}$ and ${p_{(n+1)}}$ are consecutive prime numbers, then ${p_{(n+1)}^{1/(n+1)}< p_{n}^{1/n}}$ for every ${n\geq 1}$. Rosser's inequality for the ${n}$th and ${(n+1)}$th roots, changes from strictly increasing to strictly decreasing for ${n\geq 21}$. The inequality is considered for ${n>e^{e^{3/2}}}$, i.e., ${n\geq 89}$, but since the inequalities for ${n\geq 195340>e^{e^{5/2}}}$, are also required, these inequalities are explicitly proven as well. Silva has already verified Firoozbakht's conjecture up to ${p_{n}<4 \times 10^{18}}$, and the additional theorem is proven here that there is the smallest natural number, ${m>n\geq 1}$ and ${p_{m}^{1/m}< p_{n}^{1/n}}$. It is also shown that there is a unique one to one function, which maps each element ${p_n}$ to each element ${p_{n}^{1/n}}$ for every ${n\geq 1}$ and ${1<p_{n}^{1/n}\leq 2}$. Finally, it is proved that there is a strictly decreasing sequence and Firoozbakht's conjecture would be true for all ${n\geq 1}$.