7. We recall that $\bar{E}_n$ denotes the class of $E_n = \operatorname{Card} \operatorname{MD}(n)$ in $\mathbb{Z}_p$. a. Show that $E_{2n+1} \equiv u_{2n}[p]$, where $u_m$ is the coefficient of the term $X$ in the decomposition of $\delta_{p-1}^m(X)$ in the basis $(X, \ldots, X^{p-1})$. b. Prove that the sequence $(\bar{E}_{2n+1})_{n \in \mathbb{N}}$ is periodic, with minimal period $(p-1)/2$ if $p \equiv 1\,[4]$ and minimal period $(p-1)$ if $p \equiv 3\,[4]$. c. Indicate the modifications to be made to the preceding questions to show an analogous result for the sequence $(\bar{E}_{2n})_{n \in \mathbb{N}}$.
7. We recall that $\bar{E}_n$ denotes the class of $E_n = \operatorname{Card} \operatorname{MD}(n)$ in $\mathbb{Z}_p$.\\
a. Show that $E_{2n+1} \equiv u_{2n}[p]$, where $u_m$ is the coefficient of the term $X$ in the decomposition of $\delta_{p-1}^m(X)$ in the basis $(X, \ldots, X^{p-1})$.\\
b. Prove that the sequence $(\bar{E}_{2n+1})_{n \in \mathbb{N}}$ is periodic, with minimal period $(p-1)/2$ if $p \equiv 1\,[4]$ and minimal period $(p-1)$ if $p \equiv 3\,[4]$.\\
c. Indicate the modifications to be made to the preceding questions to show an analogous result for the sequence $(\bar{E}_{2n})_{n \in \mathbb{N}}$.