counterexample

NEWS: 13532385396179

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

Screen Shot 2017-06-10 at 2.37.23 PM.png

where p is the largest prime factor of n. This motivated him to look for x of the form

Screen Shot 2017-06-10 at 2.38.05 PM.png

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.

M x

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