Exercise 6, § 2

Functions of a Real Variable, VI, p. 24

a) The rational numbers $b_n$ being defined by formula (20) of VI, p. 7, denote by $d_n$ the denominator $> 0$ of $b_n$ written in the form of an irreducible fraction. Now prove that no prime factor of $d_n$ can be $> n + 1$ (use the recurrence formula (23) of VI, p. 8). b) Now prove that one has, for every integer $p > 0$ and every integer $n > 0$

S_n(p) = b_n p + \binom{n}{1} \frac{p}{2} b_{n+1} p + \cdots + \binom{n}{r} \frac{p^r}{r+1} b_{n-r} p + \cdots + \frac{p^{n+1}}{n+1}.

c) Deduce from b) by induction on $n$ that, for every prime number $p$ the denominator of $S_n(p) - b_n p$ written in the form of an irreducible fraction, is not divisible by $p$ (observe that $p^{r+1}$ cannot divide $r + 1$ ). d) Deduce from c) that the number

b_n - \sum_p \frac{S_n(p)}{p}

where $p$ runs through the set of prime numbers $p \leq n + 1$ and $n$ is even, is an integer. Conclude that

b_{2n} + \sum_p \frac{1}{p}

where $p$ runs through the set of prime numbers such that $p - 1$ divides $2n$ , is an integer (Clausen-von Staudt theorem; use Exercise 5 e).

Solution

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