Hailstones do NOT "reign"

Numbers are fascinating. I don't mean "sheer" numbers, I mean the properties of the integers, ie whole numbers.

Twin primes, for example. Numbers like 29 and 31, numbers separated by a difference of 2, yet are prime numbers. 11 and 13 are another example of a twin prime set. There was recently a proof that the numnber of twin primes is infinite, but only recently.

Math guys are always inventing new ways to look at numbers. Twin primes is just an example.

I started this post with the intention of looking at the "persistence" of numbers, but I think I will tell about yet another invention of the mathematicians, the "hailstone" numbers.

Actually, they are not numbers at all, just the way they behave.

They way you do it is this: Select a number, any number, say, 13. The instructions are as follows.......if your selected number is even, halve it. If it is odd, like 13, triple it and add 1. For 13, this gets us to 40. Then, continue, and in this case, we have to halve it to get 20, and then again to get 10, and so on to 5. But 5 is odd, so we triple it and add 1 to get 16. This number is even again, so we halve it, to get subsequently 8, then 4, 2 and finally 1.

....and that was just 13. It took 10 "steps" to get to 1. You can do it with any number you choose.

The question is: Do ALL numbers end in 1? The math guys still do not know the answer. Some numbers take an enormous amount of steps to get to 1. Try 4173!

The reason they are called "hailstone" numbers is that the behaviour is similar to that of hailstones in a cloud, which oscillate up and down until they become heavy enough to fall to the ground, or, in our case, 1.

No-one has ever proved or disproved that all numbers end up as 1, which puts this possibly in the "Godel-like" category (see previous post). However, it is fun to try, and it not only is applicable to the "triple and add 1" rule; you can use other rules like "multiply by five and add 3" for instance.

The results are the same. ALL numbers seem to end up as 1, but there is no proof that they are infinite, unlike the twin primes.

Who is to say that someone, somewhen, somehow, will come up with a proof.

Much like Wiles did with Fermat's Last Theorem, but that is another post!