# Arithmetic Progressions of Length Three in Multiplicative Subgroups of $\mathbb{F}_p$

## TL;DR

This paper presents an efficient algorithm for detecting non-trivial three-term arithmetic progressions in multiplicative subgroups of finite fields, enabling polynomial-time computation of related combinatorial numbers.

## Contribution

The paper introduces a significantly more efficient algorithm for finding 3-APs in multiplicative subgroups, improving over naive methods.

## Key findings

- Algorithm detects 3-APs more efficiently
- Polynomial-time computation of Var der Waerden-like numbers
- Enhanced understanding of arithmetic progressions in finite fields

## Abstract

In this paper, we give an algorithm for detecting non-trivial 3-APs in multiplicative subgroups of $\mathbb{F}_p^\times$ that is substantially more efficient than the naive approach. It follows that certain Var der Waerden-like numbers can be computed in polynomial time.