On the Firoozbakht's conjecture

TL;DR

This paper proves Firoozbakht's conjecture by applying inequalities related to prime distribution, establishing the decreasing nature of the sequence of prime roots, and confirming the conjecture's validity for all natural numbers.

Contribution

The paper provides a rigorous proof of Firoozbakht's conjecture using Rosser and Schoenfeld's inequality, extending verification to all natural numbers and establishing the sequence's strict decrease.

Findings

Firoozbakht's conjecture is proven true for all n ≥ 1.

The sequence of prime roots is strictly decreasing.

A unique mapping between primes and their n-th roots is established.

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 ${n\geq 1}$.