A few years ago John H. Conway offered five $1000 problems: http://www.cheswick.com/ches/conway1000.pdf I'll transcribe the fifth one (Climb to a Prime): _____ Let n be a positive integer. Write the prime factorization in the usual way, e.g. 60 = 2^2*3*5, in which the primes are written in increasing order, and exponents of 1 are omitted. Then bring exponents down to the line and omit all multiplication signs, obtaining a number f(n). Now repeat. So, for example, f(60) = f(2^2*3*5) = 2235. Next, because 2235 = 3*5*149, it maps, under f, to 35149, and since 35149 is prime, it maps to itself. Thus 60 -> 2235 -> 35149 -> 35149 -> ..., so we have climbed to a prime, and we stop there forever. The conjecture, in which I seem to be the only believer, is that every number eventually climbs to a prime. The number 20 has not been verified to do so. Observe that 20 -> 225 -> 3252 -> 223271 -> ..., eventually getting to more than one hundred digits without yet reaching a prime! _____ A James Davis contacted me yesterday with word of a counterexample to Conway's conjecture (that every number eventually climbs to a prime) which makes him eligible (I think) for the prize. I told him to write Conway a letter. In the meantime, I'll publish his find here: 13532385396179 -> 13532385396179 -> composite forever James Davis writes: "I'm not a mathematician by any stretch - the search that happened to work was hoping n = x*p=f(x)*10^y+p, where p is the largest prime factor of n. That requires that f(x)/(x-1) terminate and it's decimal expansion be a prime (p). That motivates looking for x of the form x=m*10^y+1 and hoping some common factors cancel between f(x) and (x-1). Turned out to be enough: m=1407 fell out immediately."