Exercise 10, § 2

With the notations of Exercise 9, now prove that the numbers $m^n a_n = m^n(m^n - 1) b_n/n$ are integers (write $F_m(z)$ as a quotient of two polynomials in $e^z$). Deduce that (for $n$ pair $\geq 2$) the denominator $\delta_n$ of $b_n/n$ written in the form of an irreducible fraction is such that, for every integer $m$ prime to $n$, one has $m^n \equiv 1 \pmod{\delta_n}$. Moreover, if an integer $d$ is such that $m^n \equiv 1 \pmod{d}$ for every integer $m$ prime to $n$, $d$ divides $\delta_n$ (use the theorem of Clausen-von Staudt and the structure of the multiplicative group of $\mathbf{Z}/d\mathbf{Z}$ (A, VII, § 2, n° 4)).