Show that the product of two consecutive natural numbers is never a square. I'd like to have my proof verified and if possible, to see other solutions that are interesting.
Proof: Suppose $n(n+1)$ is a square. Then we write
$$n(n+1) = \prod_{p} p^{c(p)}$$
where $c(p) = a(p) + b(p)$ are such that
\begin{align*}
n &= \prod_p p^{a(p)} \\
n+1 &= \prod_p p^{b(p)}
\end{align*}
Now, by our hypothesis, $c(p)$ is even for all primes $p$. As $(n,n+1)=1$ for all $n$, it must be that $a(p)$ and $b(p)$ are even for all primes $p$ and moreover, $a(p) = 0$ whenever $b(p)>0$ and reversely.
This indicates that both $n$ and $n+1$ are squares. This is impossible as there are no consecutive squares in the natural numbers.
 A: $$
n^2<n(n+1)<(n+1)^2
$$
That's all :)
There are no integer number between $n$ and $n+1$.
A: It looks like this may be the same proof as the OP, but here's how I'd write it:
Let $n$ and $n+1$ be consecutive natural numbers.  Note that they must have disjoint prime factorizations.  I.e. if $p|n$, then $p \nmid (n+1)$.  
For $n(n+1)$ to be a perfect square, then the power on every prime in its decomposition must be even.  However, since $n$ and $n+1$ have disjoint prime decompositions, then this is only possible if the power on every prime in the decomposition of $n$ is even, and similarly for $n+1$.  This would imply that both $n$ and $n+1$ are perfect squares, which is impossible since no two consecutive naturals can be perfect squares.

Granted, this proof is a lot more clumsy than Hagen's and Oleg's.
A: This is fine if $n=0$ is not allowed (and if $n=0$ is allowed the claim becomes wrong.
As you ask for alternative proofs, what can you see from this inequality:
$$ n^2<n^2+n<n^2+2n+1$$
A: Well I have another explain for it, but I'm not sure if it's true or not, So I will be happy that somebody correct me :)
We want to discuss about if n(n+1) is square or not, so we try to make it square:
$n(n+1) = n^2 + n = n^2 + n + \frac 14 - \frac 14 = (n+\frac12)^2 - \frac 14$ 
Now let X = $(n+\frac12)$
Now we have:
$$ n(n+1) = X^2 - \frac 14$$
So as you can see, this equation shows us that $n(n+1)$ is always not equal to a square.
