As detailed here, if you consider “the behavior of the iterates of the function which takes odd integers n to 3n+1 and even integers n to n/2, the ‘3x+1 Conjecture’ asserts that, starting from any positive integer n, repeated iteration of this function eventually produces the value 1.” There are actually lots of sites which address this function (e.g. here).
I don’t have a clue about 99% of the contents of these sites, but I do have a simple question:
Are there are other, similar, functions on the integers that ultimately produce the value 1?
Certainly. There are an infinite number of functions which converge to 1. If x[sub]n[/sub] is a series such that x[sub]n[/sub] > x[sub]n+1[/sub] and x[sub]n[/sub] > 1 but x[sub]n+1[/sub] is not bounded below by any greater constant, then any function that (in general) takes x[sub]n[/sub] to x[sub]n+1[/sub] converges to 1.
More on this tomorrow–I’ll have time to look up theorems then.
Thanks ultrafilter, but I think you’re giving me too much credit. I’m just looking for another “rule” such as if you take odd integers to k[sub]1[/sub]n+k[sub]2[/sub] and even integers to n/2, then you’ll always arrive at 1.
What’s interesting about this function (which I know of as the hailstone problem for what are obvious reasons if you actually watch its behavior) is that no one knows if it always hits the 4,2,1 cycle, although lots of people have worked on it. In that sense I know of no similar functions.