A note on the decimal expansion of reciprocals of Mersenne primes (Q6542775)

The aim of this note is to study, in the realm of Midy's theorem, arithmetic properties of digital blocks in the base-10 expansion of \(1 / M_p\), with \(M_p=2^p-1\) a Mersenne prime. If \(p\) divides the period length \(L\) of \(1/M_p\), one can partition the periodic part into \(p\) blocks, each of length \(\ell\). The author shows that there exists a permutation \(B_1^{\prime}, \ldots, B_p^{\prime}\) of these blocks such that\N\begin{itemize}\N\item[(i)] \(B_1^{\prime}=2 B_p^{\prime}-10^{\ell}\), \(B_{k+1}^{\prime}=2 B_k^{\prime}\) for \(1 \leq k \leq p-2, B_p^{\prime}=2 B_{p-1}^{\prime}+1\), and\N\item[(ii)] \(B_1^{\prime}+\cdots+B_p^{\prime}=10^{\ell}-1\).\N\end{itemize}\NThe proof is elementary. Some data and examples are listed, too.