# A Note on the Bateman-Horn Conjecture

## TL;DR

This paper empirically investigates the Bateman-Horn conjecture, identifying deviations with non-monic polynomials and proposing a modified version that improves accuracy for smaller prime values.

## Contribution

It introduces a modified form of the Bateman-Horn conjecture that better aligns with empirical data, especially for non-monic polynomials and small primes.

## Key findings

- Original conjecture shows large deviations with non-monic polynomials.
- Modified conjecture demonstrates improved empirical accuracy.
- Enhanced predictive power for small prime values.

## Abstract

We report the results of our empirical investigations on the Bateman-Horn conjecture. This conjecture, in its commonly known form, produces rather large deviations when the polynomials involved are not monic. We propose a modified version of the conjecture which empirically demonstrates remarkable accuracy even for modest values of primes.