I discovered this by accident when I first learned about continued fractions. It's hardly foolproof, but maybe half the time, you can instantly factor semiprimes if you have one in continued fraction form.
Example:
$$\sqrt{47 \cdot 97} = [67; 1, 1, 11, 1, 3, 2, 3, 2, 2, 2, 3, 2, 3, 1, 11, 1, 1, 134].$$
Then you swap the first half of the range with the second half of the range and change the leading term to be half the final term, like so, sort of turning the whole thing inside-out:
$$[1; 2, 3, 2, 3, 1, 11, 1, 1, 134, 1, 1, 11, 1, 3, 2, 3, 2, 2].$$
And then you convert it back to its radical form, and square it.
What does this give you? Well, when it works, as it does here, you get
$$\frac{97}{47},$$
a result which is especially impressive when you're dealing with giant terms. Of course, the number of terms in continued fractions makes this impractical out past $10^{40}$ or something like that.
Anyway, does anyone know why this works$-$and why it doesn't work some of the time? Sometimes the factors split nicely like this, and sometimes they remain together.