Exercise 10, § 2

With the notation of exerc. 9 show that the numbers $m^n a_n = m^n(m^n - 1)b_n/n$ are integers (write $F_m(z)$ as the quotient of two polynomials in $e^z$). Deduce that (for n even $\geqslant 2$) the denominator $\delta_n$ of $b_n/n$ in irreducible form is such that, for every integer m prime to n one has $m^n \equiv 1$ (mod. $\delta_n$). Further, if an integer d is such that $m^n \equiv 1$ (mod. d) for every integer m prime to n, then d divides $\delta_n$ (use the Clausen-von Staudt theorem and the structure of the multiplicative group of $\mathbf{Z}/d\mathbf{Z}$ (Alg., VII. 12)).