Addendum. Since I had already TeXed Witt's proof for my Notes, it is easy to reproduce it here :

Theorem (von Staudt--Clausen, 1840)
Let $k>0$ be an even integer, and let $l$ run through the primes. Then the(1)\quad\quad\quad W_k=B_k+\sum_{l-1|k}{1\over l}is always an integer. For example, $\displaystyleW_{12}=B_{12}+{1\over 2}+{1\over3}+{1\over5}+{1\over7}+{1\over 13}=1$.

[The British analyst Hardy says in his Twelve lectures (p. 11) that this theorem was rediscovered by Ramanujan at a time of his life when he had hardly formed any definite concept of proof''.]

Proof (Witt) : The idea is to show that $W_k$ is a $p$-adic integerfor every prime $p$. More precisely, we show that $B_k+p^{-1}$ (resp. $B_k$)is a $p$-adic integer if $p-1|k$ (resp. if not).

For an integer $n>0$, let $S_k(n)=0^k+1^k+2^k+\cdots+(n-1)^k$. Comparing thecoefficients on the two sides of1+e^T+e^{2T}+\cdots+e^{(n-1)T}={e^{nT}-1\over T}{T\over e^T-1},we get $\displaystyle S_k(n) =\sum_{m\in[0,k]}{k\choose m}{B_m\over k+1-m}n^{k+1-m}$. To recover $B_k$ from the $S_k(n)$, it is tempting totake the limit $\displaystyle\lim_{n\to0}S_k(n)/n$, which doesn't make sensein the archimedean world. If, however, we make $n$ run through the powers$p^s$ of a fixed prime $p$, then, $p$-adically, $p^s\to0$ as $s\to+\infty$,Let us compare $S_k(p^{s+1})/p^{s+1}$ with $S_k(p^s)/p^s$. Every$j\in[0,p^{s+1}[$ can be uniquely written as $j=up^s+v$, where $u\in[0,p[$ and$v\in[0,p^s[$. Now,% &\equiv p\left(\sum_{v\in[0,p^s[}v^k\right)% +kp^s\left(\sum_{u\in[0,p[}u\sum_{v\in[0,p^s[}v^{k-1}\right)% \pmod{p^{2s}}\cr
&\equiv p\left(\sum_v v^k\right) +kp^s\left(\sum_u u\sum_v v^{k-1}\right)\pmod{p^{2s}}\cr by the binomial theorem. As $\sum_{v}v^k=S_k(p^s)$ and$2\sum_uu=p(p-1)\equiv0\pmod p$, we getS_k(p^{s+1})\equiv pS_k(p^s)\pmod{p^{s+1}},where, for $p=2$, the fact that $k$ is even has been used. Dividingthroughout by $p^{s+1}$, this can be expressed by saying that{S_k(p^{s+1})\over p^{s+1}}-{S_k(p^s)\over p^s}\in{\mathbf Z}_{(p)}is a $p$-adic integer, and therefore{S_k(p^r)\over p^r}-{S_k(p^s)\over p^s}\in Z_{(p)}for any two integers $r>0$, $s>0$, since ${\mathbf Z}_{(p)}$ is a subring of $\mathbf Q$.Fixing $s=1$ and letting $r\to+\infty$, we see that $B_k-S_k(p)/p\in{\mathbf Z}_{(p)}$,in view of $(2)$. We need a

Lemma.$S_k(p)=\sum_{j\in[1,p[}j^k$ is $\equiv-1\pmod p$ if $p-1|k$ and $\equiv0\pmod p$ otherwise.

This is clear if $p-1|k$. Suppose not, and let $g$ be a generator of$({\mathbf Z}/p{\mathbf Z})^\times$. We have $g^k-1\not\equiv0$, whereas\equiv (g^k-1)\left(\sum_{t\in[0,p-1[}g^{tk}\right)\equiv g^{(p-1)k}-1\equiv0.It follows that $B_k+p^{-1}\in{\mathbf Z}_{(p)}$ if $p-1|k$ and $B_k\in{\mathbf Z}_{(p)}$otherwise. In either case, the number $W_k$ (1), which can be written as(B_k+p^{-1})+\sum_{l\neq p}l^{-1}&\hbox{if }p-1|k\cr(B_k)+\sum_l l^{-1}&\hbox{otherwise},\cr}$$(where l runs through the primes for which l-1|k) turns out to be ap-adic integer for every prime p. Hence W_k\in{\mathbf Z}, as claimed. 2 added 52 characters in body; edited body Here are three more elementary results whose most natural proof involves p-adic ideas. All three can be found in Cassels' book on Local Fields (edit mentioned by Laurent Berger in a comment). The first one is Witt's proof of the theorem of Clausen and von Staudt in Chapter 1. It requires nothing more than the definition of the p-adic valuations; the idea is that a\in{\bf Q} is in {\bf Z} if a\in{\bf Z}_p for every prime p. The second one says that the order of every finite subgroup G\subset{\mathrm GL}_n({\mathbf Q}) divides$$ \prod_l l^{\beta(l)} $$where l runs over the primes and$$\beta(l)=\lfloor n/(l-1)\rfloor+\lfloor n/l(l-1)\rfloor+\lfloor n/l^2(l-1)\rfloor+\cdots $$for l\neq2 and \beta(2)=n+2\lfloor n/2\rfloor+\lfloor n/2^2\rfloor+\lfloor n/2^3\rfloor+\cdots. See Theorem 2.1 in Chapter 4. The third one is a theorem of Selberg which says that every finitely generated subgroup G\subset{\mathrm GL}_n(k), where k is a field of characteristic 0, contains a normal torsionfree subgroup of finite index. See Theorem 4.1 in Chapter 5. Note finally that the (Skolem)-Mahler-Lech theorem is Theorem 5.1 in the same chapter. 1 Here are three more elementary results whose most natural proof involves p-adic ideas. All three can be found in Cassels' book on Local Fields. The first one is Witt's proof of the theorem of Clausen and von Staudt in Chapter 1. It requires nothing more than the definition of the p-adic valuations; the idea is that a\in{\bf Q} is in {\bf Z} if a\in{\bf Z}_p for every prime p. The second one says that the order of every finite subgroup G\subset{\mathrm GL}_n({\mathbf Q}) divides$$ \prod_l l^{\beta(l)} $$where l runs over the primes and$$\beta(l)=\lfloor n/(l-1)\rfloor+\lfloor n/l(l-1)\rfloor+\lfloor n/l^2(l-1)\rfloor+\cdots  for $l\neq2$ and $\beta(2)=n+2\lfloor n/2\rfloor+\lfloor n/2^2\rfloor+\lfloor n/2^3\rfloor+\cdots$. See Theorem 2.1 in Chapter 4.

The third one is a theorem of Selberg which says that every finitely generated subgroup $G\subset{\mathrm GL}_n(k)$, where $k$ is a field of characteristic $0$, contains a normal torsionfree subgroup of finite index. See Theorem 4.1 in Chapter 5.

Note finally that the (Skolem)-Mahler-Lech theorem is Theorem 5.1 in the same chapter.
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