Numbers don't ever End

Way back in the Anal(ls) of this blog I referred to the "persistence" of numbers

look up "Hailstone Numbers")

It goes something like this. A number's "persistence" is defined as how long it can keep going if you multiply its digits together...for example, take the number 213.

213...2 x 3 x 1 = 6 , persistence is 1, one step to 1 digit.

2132 = 2 x 1 x 3 x 2 = 12, then 12 = 1 x 2 = 2 , persistence is 2 ( two steps to get to 1 digit). Make it up as you go along.

Try it! Take that number, 4174.....4 x 7 x 1 x 4 = 112
1 x 1 x 2 = 2..persistence 2
Try 4173......4 x 1 x 7 x 3 = 84, 8 x 4 = 32, 3 x 2 = 6, so ...

4174 has persistence 2
4173 has persistence 3

Try and find a number with persistence 5. (clue - it's between 600 and 6000)

That is just "multiplicative" persistence.
Try "additive" persistence!

What is the smallest number of additive persistence of 6? Or multiplicative persistence of 6?

The conjecture is that no number EVER has persistence greater than 11, yet another example of Godel's Theorem - ie it is neither provable nor disprovable.