# Generalizations of an Ancient Greek Inequality about the Sequence of Primes

## TL;DR

This paper extends an ancient Greek inequality on prime sequences to arithmetic progressions and polynomial cases, refines related conjectures, and offers insights on ongoing conjectures in prime number theory.

## Contribution

It generalizes a classical prime inequality to broader polynomial and progression contexts and refines existing conjectures in prime distribution.

## Key findings

- Generalized prime inequalities to arithmetic progressions and multivariable polynomials
- Refined Bouniakowsky's conjecture and related conjectures
- Provided remarks on conjectures in recent prime number research

## Abstract

In this note, we generalize an ancient Greek inequality about the sequence of primes to the cases of arithmetic progressions even multivariable polynomials with integral coefficients. We also refine Bouniakowsky's conjecture [16] and Conjecture 2 in [22]. Moreover, we give two remarks on conjectures in [22]