# Relaxing the size constraints on the criterion of Proth

## TL;DR

This paper extends Proth's primality test to larger numbers by adding a simple, efficient condition that preserves the test's biconditionality and can be computed concurrently, broadening its applicability.

## Contribution

The authors introduce a new condition to Proth's theorem, allowing it to test a wider class of numbers without added complexity, and discuss an extension to an existing primality test.

## Key findings

- Extended Proth's theorem to larger numbers
- Maintained the theorem's biconditionality
- Added minimal computational complexity

## Abstract

We add one condition to the theorem of Proth to extend its applicability to $N=k2^n+1$ where $2^n>N^{1/3}$ as opposed to the former constraint of $2^n>k$. This additional condition adds barely any complexity or time to the test and can furthermore be calculated concurrently. Furthermore, it maintains the biconditionality of the theorem and thus makes it readily applicable. A note on an extension of the primality test of Brillhart, Lehmer, and Selfridge is also made.