Since the last issue of Parabola Incorporating Function the mathematics world has been surprised by the appearance of a preprint, dated May 2011, by the German mathematician Gerhard Opfer who claimed to have proved the Collatz Conjecture.  And then, some weeks later, other mathematicians reported that the proof was flawed and Opfer updated his preprint on 17 June 2011 with ``the statement `that the Collatz conjecture is true' has to be withdrawn, at least temporarily". The Collatz Conjecture is one of those remarkable mathematical problems that is easy to state and easy to check but difficult to prove. Starting with any natural number $n_0>1$, multiply the number by three and then add one if it is odd, or divide the number by two if it is even to construct another natural number $n_1$. Then apply the same rule to $n_1$ to construct another natural number $n_2$ and so on, but stop if you ever reach the number 1. The Collatz Conjecture says that the sequence $n_0, n_1, n_2, \ldots$ will always reach the number 1. Try it out, suppose we start with seven, then we obtain the sequence:
$$7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1$$
The Wikipedia page on the Collatz Conjecture is very informative.