20

This number is a composite.

Let n be a positive integer. Write the prime factorization in canonical (standard) form. For example, 120 = 2 ^ 3 * 3 * 5 where the primes are written in increasing order of magnitude, and exponents of 1 are omitted. We then bring exponents down to the line and omit all multiplication signs, obtaining a number f(n). Now repeat. So, for example, f(120) = f(2 ^ 3 * 3 * 5) = 2335. Next, because 2335 = 5 * 467, it maps, under f, to 5467. Now f(5467) = f(7 *11 *71) = 71171. Since 71171 is prime, it maps to itself. Thus 120 → 2335 → 71171→...,so we have climbed to a prime, and we stop there forever. 20 is the first composite integer where this iterative factorization and concatenation procedure does not yet yield a prime after getting to more than one hundred digits. The initial ten iterations for 20 are 20 → 225 → 3252 → 223271 → 297699 → 399233 → 715623 → 3263907 → 32347303 → 160720129 → .... This was the fifth problem in John Horton Conway's list of prize problems. James Davis in 2017 proved that the Climb to a Prime conjecture was indeed false. The problem is a neat variation of The Home Prime Conjecture which remains open. [Schiffman]