Alex's Adventures in Numberland - Alex Bellos [108]
Through the Encyclopedia, Sloane sees a lot of new mathematical ideas, and he also spends time inventing his own. In 1973 he came up with the concept of the ‘persistence’ of a number. This is the number of steps that it takes to get to a single digit by multiplying all the digits of the preceding number to obtain a second number, then multiplying all the digits of that number to get a third number, and so on until you get down to a single digit. For example:
88 8 × 8 = 64 6 × 4 = 24 2 × 4 = 8
So, according to Sloane’s system, 88 has persistence 3, since it takes three steps to get to a single digit. It would seem likely that the bigger a number is, the bigger its persistence. For example, 679 has persistence 5:
679 378 168 48 32 6
Likewise, if we worked it out here, we would find that 277777788888899 has persistence 11. Yet here’s the thing: Sloane has never discovered a number that has a persistence greater than 11, even after checking every number all the way up to 10233, which is 1 followed by 233 zeros. In other words, whatever 233-digit number you choose, if you follow the steps of multiplying all the digits together according to the rules for persistence, you will get to a single-digit number in 11 steps or fewer.
This is splendidly counter-intuitive. It would seem to follow that if you have a number with 200 or so digits consisting of lots of high digits, say 8s and 9s, then the product of these individual digits would be sufficiently large that it would take well over 11 steps to reduce to a single digit. Large numbers, however, collapse under their own weight. This is because if a zero ever appears in the number, the product of all the digits is zero. If there are no zeros in the number to start with, a zero will always appear by the eleventh step, unless the number has already been reduced to a single digit by then. In persistence Sloane found a wonderfully efficient giant-killer.
Not stopping there, Sloane has compiled the sequence in which the nth term is the smallest number with persistence n. (We are considering only numbers with at least two digits.) The first such term is 10, since:
10 0 and 10 is the smallest two-digit number that reduces in one step.
The second term is 25, since:
25 10 0 and 25 is the smallest number that reduces in two steps.
The third term is 39, since:
39 27 14 4 and 39 is the smallest number that reduces in three steps.
The full list is:
(A3001) 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899
I find this list of numbers strangely fascinating. There is a distinct order to them, yet they also are a bit of an asymmetric jumble. Persistence is sort of like a sausage machine that produces only 11 very curiously shaped sausages.
Sloane’s good friend Princeton professor John Horton Conway also likes to amuse himself by coming up with offbeat mathematical concepts. In 2007 he invented the concept of a powertrain. For any number written abcd…, its powertrain is abcd…In the case of numbers where there is an odd number of digits, the last digit has no exponent, so abcde goes to abcde. Take 3462. It reduces to 3462 = 81 × 36 = 2916. Reapply the powertrain until only a single digit is left:
3462 2916 2916 = 512 × 1 = 512 512 = 10 10 = 1
Conway wanted to know if there were any indestructible digits – numbers that did not reduce to a single digit under the powertrain. He could find only one:
2592 2592 = 32 × 81 = 2592
Not one to sit idly by, Neil Sloane took up the chase and uncovered a second:*
24547284284866560000000000
Sloane is now confident that there are no other indestructible digits.
Consider that for a moment: Conway’s powertrain is such a lethal