Solving the Fundamental Equation \(y^{\bar{\tau}(r)} = y\) in \(\mathbb{Z}_r\)

In this short note, we provide a compact formula for the smallest (necessarily odd) integer \(\bar{\tau}(r) > 1\) such that the fixed-point equation \(y^{\bar{\tau}(r)} = y\) admits all solutions arising from the family of equations \(y^{\tau}=y\) with \(\tau \geq 2\) in the commutative ring of \(r\)-adic integers.