Define a map f as follows: if abcd … is the decimal representation of the integer n, then f (n) is the product *a b × c d... When n has an odd number of digits, the final digit in f(n*) has no exponent, and by convention we set 00 = 1.
Thus, *f (2564) = 25 × 64 = 41,472 and *f (23417) = 23 × 41 × 7 = 224. Iterating this map always leads to single-digit numbers, which are fixed points of f.
Thus, f (2564) = 41,472, f (41,472) = 131,072 and *f (131,072) = 0.
Similarly, f (23417) = 224, f (224) = 16 and *f (16) = 1. In fact, this is not always true! There are exceptions: two other fixed points are known. It is conjectured that there are no others. Searches have been carried out up to 10100.
These two fixed points are: 2592 = 25 × 92 and
24,547,284,284,866,560,000,000,000 = 24 × 54 × 72 × 84 × 28 × 48 × 66 × 56, since recall that 00 = 1.
These fixed points can also be reached:
for example, f (45,756) = 103,262,208 and f (103,262,208) = 2592.
John Conway (see "The genius of John Conway", Tangente no. 194, 2020) introduced this map f and its iterates in 2007. He called them power trains.
In 2014, he also introduced another map, which is in some sense its inverse. Denoted by g, it is defined as follows: if the prime factorization of an integer n is *ab × c d…, then g(n) is abcd … For example, g(12) = g* (22 × 3) = 223. Every prime number, along with 1, is a fixed point of g. We can also iterate the map g. In the previous example, the iteration stops because 223 is prime, so g (223) = 223. Starting from 9 = 32, we successively obtain 32, 25 and 52, before stopping at 2213, which is prime. Conway conjectured that the iteration always ends at a prime number.
But in 2017, James Davis found a fixed point that is not prime.
The number 13,532,385,396,179 = 13 × 532 × 3853 × 96179 is a fixed point! The conjecture is therefore false.
