i wrote

if 0 is not in D , then it's possible to get elements of S that end with 0 . this may not always occur, but it seems likely if the set S is reasonably large. the rightmost 0's double with each squaring and do not affect the other digits. thus, keeping the rightmost part of the splitting, we get arbitrarily large numbers, unless a splitting digit occurs as the rightmost non-zero digit.

the rightmost non-zero digit converges to an idempotent modulo b (the base). so if all of these idempotent are in the splitting set D , that prevents this process from producing arbitrarily large numbers. the first interesting case appears to be b = 3 , D = { 1 } . i've tried several choices for the initial number in S , and so far all have yielded only finitely many elements in S . in this case, elements in S have only 0's and 2's in their ternary representations ("cantor numbers"??), except perhaps the starting number. from the squares of cantor numbers, we excise 0-2 strings, and can safely eliminate any rightmost 0's because they'll get removed at the next iteration. most of the resulting numbers seem to be smaller than the previous number, but there are exceptions, such as 2 0^n 2 0^{n+1} 2 (where exponents indicate that many repetitions) whose square produces this palindromic substring: 22 0^{n-1} 202 0^{n-1} 22 is there an initial number for which S is infinite in this case (b = 3, D = {1}) ? rich: ) > do you know of any examples where 0 is not in D , and you can prove ) > that S is infinite? ) Sure. Base 4, D = {3}, starting value {11001}. ) The right ends of successive squarings will be xxx11[0*]1 and xxx22[0*]1. ) This is an infinite set, as the string of internal 0s grows by one 0 with ) each 4th-powering. this generalizes to any base that's a power of 2 . any other examples where we can show that S is infinite? (a trivial one is to take the set of splitting digits to be empty.) another question: is there an example of an initial number A , and a base b , and set of splitting digits D , such that the set S is finite, but for the same A and b , but for a larger set D , the set S is infinite? too many questions, not enough answers! mike