Recently, James Davis found a counterexample to John H. Conway’s ‘Climb to a Prime’ conjecture, for which Conway was offering $1,000 for a solution.
The conjecture states the following:
“Let n be a positive integer. Write the prime factorisation in the usual way, where the primes are written in ascending order and exponents of 1 are omitted. Then bring the exponents down to the line, omit the multiplication signs, giving a number f(n). Now repeat.”
For example, f(60) = f(2^2 x 3 x 5) = 2235. As 2235 = 3 x 5 x 149, f(2235) = 35149. Since 35149 is prime, we stop there.
Davis had a feeling that the counterexample would be of the form
where p is the largest prime factor of n. This motivated him to look for x of the form
The number Davis found was 13532385396179 = 13 x 53^2 x 3853 x 96179, which maps to itself under f (i.e. its a fixed point). So, f will never map this composite number to a prime, hence disproving the conjecture.