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.

To generate the next number in the sequence from the previous term, read off the digits of the previous number, counting the number of digits in groups of the same digit. For example:

1 is read as ‘one 1’ = 11

11 is read as ‘two 1s’ = 21

21 is read as ‘one 2, one 1’ = 1211

If we start with any digit x from 0 to 9, then x will remain the last digit of the sequence. When x does not equal 1, the sequence is as follows:

The Conway sequence, named by Vardi in 1991, is a look-and-say sequence with the starting digit 3.

Growth in Length and Conway’s Constant

The terms of the sequence eventually grow in length about 30% per generation. If L_{n} denotes the number of digits in the n-th term of the sequence, the limit of the ratio

is Conway’s constant:

where λ = 1.303577269034…

Conway’s constant is the unique positive real root of the following polynomial:

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