In The Hailstone Tree, it was concluded that finding a number which is not on the Hailstone Tree would disprove the Collatz conjecture. The properties that such a number would have are interesting to consider. At first, let us consider the smallest such number. It will of course be an odd number. If it were not we could simply divide by two to get a smaller number which also would not be on the tree. Being an odd number it would be the first number of a structure exactly analogous to a branch on the tree, except that it would in no way be connected to the tree. In addition to the infinite series of even numbers which constitute the branch, it would be connected by the multiply by three and add one Hailstone rule to an even number on a lower level branch, just as happens within the tree. This leads inevitably to the appearance of a network structure made up of branches which may in fact be another tree (if it encounters a loop), or it may take some other form.

Well, number theory researchers have tested every number up to 10¹⁸ and found that they all degenerate to 1, which means that they are all included on the Hailstone Tree. Now half of these numbers are odd numbers, and so each of them marks the beginning of another new branch. This means that our hypothetical unconnected odd number finds itself in a very crowded world. A minimum of five hundred thousand trillion numbers smaller than itself are already attached to the tree, and their number is growing rapidly. This is the domain in which any kind of computer search would have to be conducted. The odds are daunting to say the least, but if anybody would like to try, the Rule of 8 function of the Hailstone Program is exactly what you need to give it a try. Set the Hailstone Options to 100 digits (or as many more as you like if you have a nice fast computer) and set the How many Numbers option to a suitably large number to keep you computer busy while you sleep. Then execute the Rule of 8 function.

In the unlikely event that it does, it will present you with a web page which will announce the happy news, and politely request that you carry out some simple actions which will result in details of the event being emailed to me. I don't expect that this will ever actually happen, but I wish you the very best of luck if you are motivated to try.