Multiplicative Persistence
277,777,788,888,899 is an unusual and special number. When it comes to Multiplicative Persistence, it is an unparalleled superstar.
Check out the following table. Can you figure out what is going on? The digits of any given number are multiplied together to get a new number, and then those digits are multiplied together, and so on. The Multiplicative Persistence of a number is equal to the number of steps required to get to a single digit. Really simple and pretty cool, isn’t it?
MP n
0 0
1 10→0
2 25→10→0
3 39→27→14→4
4 77→49→36→18→8
5 679→378→168→48→32→6
6 6788→2688→768→336→54→20→0
7 68889→27648→2688→…→0
8 2677889→338688→27648→…→0
9 26888999→4478976→338688→…→0
10 3778888999→438939648→4478976→…→0
11 277777788888899→4996238671872→438939648→…→0
So, 77 has a Multiplicative Persistence of 4 because it takes four steps to get to a single digit, in this case, 8. What about 11? Why did we stop there? Because 11 is the record, and 277777788888899 (commas aren’t necessary, right?) is the shortest number to share in that record. Other numbers, with many more digits, tie our special number, but none beat it. Did you get that? Do you fully understand the strength of that statement? The conjecture is that any number, any single one you can think of, has a Multiplicative Persistence of 11 or smaller. No number has been found that takes even 12 steps to get to a single digit.
This is quite extraordinary, don’t you think? If you like, take out a computer and start coding. Mathematical immortality awaits, but my guess is the search is futile, just like it is with the 10,958 problem I wrote about some time ago. I think that a series of digits with a Multiplicative Persistence of 12 or greater, if it exists, would have been found a long time ago.
I am happy that I get to mention the great Paul Erdos before I close out this short post. Erdos had a finger or two in this particular mathematical pie. He suggested that we ignore all zeroes and just multiply together all the other digits. After all, if you come across a zero, you are sunk. This makes for a tasty mathematical stew. There are people actively doing research in this area. If you ignore zeroes, I have seen a Multiplicative Persistence as high as 22. That said, I recently came across a paper on this very topic in French. I tried to understand it as best I could. The strange thing is that this is the first time I can remember that the specialized math did not lose me, I got lost in the language differences long before that could happen. Those French, it’s like they have a different word for everything. And yes, I tried to translate the page, but Google only decoded the numbers…
