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

##### Aseem Raj Baranwal, Jeffrey Shallit

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.

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