Persistence, Palindromes and 196

Here is a long-promised post - the "persistence of the intergers"!!!

"Peristence" is easy to define - just take a number and multiply all its digits together and continue until you are left with just one.

Example - 39. 39 becomes 3 x 9 = 27, then 27 becomes 2 x 7 = 14, and 14 becomes 1 x 4 = 4. Three steps, so the "persistence" of 39 is 3.

You can do it with any number you like. Obviously, numbers with a zero in have persistence 1, so they are not very interesting. Try finding the persistence of 6,788.

What IS interesting though is finding the SMALLEST numbers of a certain persistence. 39 is an example, and is the smallest number of persistence 3. 6,788 is the smallest of persistence 6.

That is, all numbers lower than 39 have persistence 2 or 1, and all higher than 39 have peersistence 3 or higher. For 6,788, all numbers LOWER than 6,788 have persistence 5 or less, whilst all higher have pertistence 6 or more.

Now, here's where it gets really interesting, because it is widely believed that there is absolutely NO number whatsoever with a peristence higher than 11. That, is, no number has been found to take more than 11 steps to reduce to one digit. I find this staggeriing, simply because you can invent any number as long as you like, say with 500 digits, and when you apply the persistence rule, it reduces to one digit in less than 11 steps. Try it!

Incidentally, this is just multiplicative persistence. The other one is additive persistence, where you add instead of multiply the digits, but is not quite as interesting.

Like I have said before, numbers facinate me. On a related topic, there are the palindromes, numbers like 121 and 3443 that read the same forward and backwards. If you take any number, and add it to its reversal, this produces another number. Do it again and again until a palindrome is produced. Example: 456: 456 + 654 = 1110: 1110 + 0111 = 1221 a palindrome. Try it with any number - you will eventually, after only at most 24 reversals, get a palindrome,

BUT NOT FOR 196 !!! Despite an enormous amount of computer power, 196 has still not produced a palindrome.

Nobody has the foggiest idea why 196 is special this way, but it is deeply fascinating to speculate why it is 196 and not 197 or 195, say. Try them! They both produce palindromes easily!

Watch this space for the emirps, Kaprekar numbers, and the Weird numbers!