Brady's Theorem

This paper introduces a universal invariant for repeating base‑\(b\) expansions of fractions \(a/p\) that dramatically extends the classical Midy phenomenon. For any prime \(p\), any base \(b\) with \(\gcd(b,p)=1\), and any divisor \(k\) of the period \(r\) of \(1/p\), each fraction \(a/p\) is split into \(m=r/k\) chunks of \(k\) digits, and the normalized chunk‑sum \(M_a\) is formed by dividing the chunk sum by the repunit \(b^k-1\). The main theorem proves that these values satisfy the global identity \sum_{a=1}^{p-1} M_a=\frac{(p-1)m}{2}, a relation independent of the base, the digits, or the repunit. The proof uses a pairing symmetry \(M_a+M_{p-a}=m\) arising from digitwise complements. The result generalizes Midy’s theorem to all fractions, all chunk sizes, and all bases, and reveals a deeper group‑theoretic structure connected to discrete logarithms. When \(b\) is a primitive root modulo \(p\), computational evidence suggests a striking two‑value distribution for the individual \(M_a\), leading to an open conjecture and several further research directions.