On solutions of some of unsolved problems in number theory, specifically on the distribution of primes

TL;DR

This paper claims to solve several famous unsolved problems in number theory related to prime distribution, utilizing recent proofs and theorems to establish these longstanding conjectures.

Contribution

The paper presents proofs for multiple renowned prime distribution conjectures using recent advances like Firoozbakht's conjecture and Kourbatov's theorem.

Findings

Proof of Legendre's conjecture

Proof of Cramér's conjecture

Resolution of Shanks' conjecture

Abstract

We solve some famous conjectures on the distribution of primes. These conjectures are to be listed as Legendre's, Andrica's, Oppermann's, Brocard's, Cram\'{e}r's, Shanks', and five Smarandache's conjectures. We make use of both Firoozbakht's conjecture (which recently proved by the author) and Kourbatov's theorem on the distribution of and gaps between consecutive primes. These latter conjecture and theorem play an essential role in our methods for proving these famous conjectures. In order to prove Shanks' conjecture, we make use of Panaitopol's asymptotic formula for $\pi(x)$ as well.