# Critical exponent of infinite balanced words via the Pell number system

## TL;DR

This paper proves that the minimal critical exponent of infinite balanced words over a 5-letter alphabet is 3/2, using logic, Pell number system, and automata-based computation.

## Contribution

It establishes the exact minimal critical exponent for infinite balanced words over a 5-letter alphabet, confirming a recent conjecture.

## Key findings

- Critical exponent of 3/2 for 5-letter alphabet
- Use of Pell number system in combinatorics on words
- Automata-based proof technique

## Abstract

In a recent paper of Rampersad et al., the authors conjectured that the smallest possible critical exponent of an infinite balanced word over a 5-letter alphabet is $3/2$. We prove this result, using a formulation of first-order logic, the Pell number system, and a machine computation based on finite-state automata.