joshuazelinsky | Rivers of numbers

Define a function f(x) which takes a number and adds to that number the sum of its digits in base 10. For example, f(112) = 116 because 116 = 112+1+1+2. We can keep applying this function to itself and get a sequence. For example, if we started with 7, we would get 7, 14, 19, 29, 40, 44, 52, 59... and so on.

We'll call such a sequence a river. One thing to note is that some river can start off with one number and eventually turn into another river. For example, 3,6, 12, 15, 21, 24, 48, 60, 66, 78, 93, 105, 111, 114, and 30, 33, 39, 51, 57, 69, 84, 96, 111, 114... If two rivers meet, they are then the same at any subsequence point.

Rivers are related to self-numbers which are numbers which are not in the range of f(x). They can be thought of as numbers which only ever appear at the start of a river, never in the middle.

I was introduced to the idea of rivers by this Mathoverflow question . They define three "main rivers" , 1, 2, 4, 8, 16, 23..., 3,6, 12, 15, 21..., and 9, 18, 27,... and prove that these three can never intersect (this is a good exercise!) . They ask if every river eventually meets one of these three main rivers.

However, it isn't obvious to me that there is even a finite set of rivers such that any river eventually meets one of those. So, let's do what mathematicians always do when we can't answer a question, try to ask a related question. We can define rivers the same way for any base b, where our function f_b(x) adds x to the sum of the base b digits. It isn't obvious to me how to even prove that there is any base b where there's a finite collection of rivers which are always eventually met. A small amount of playing around suggests that in base 3, every river eventually meets the river started by 1, but I don't see any obvious way to prove it.