John H. Conway is a mathematician who created these five $\$1000$ problems. They are just problems that he discovered himself and that relate to mathematics, mainly on a recreational level. Anyone to have "solved" one of the problems (by "solved", I mean provide a proof or counterexample, etc) will win $\$1000$, and will win that for every problem solved.
I looked at Problem $5$ in particular, and someone had found a counterexample to the proposition and won the prize. You can go here to check out the problems, and Problem $5$ is the very last one.
It mentions that $20$ seems to (also) be a counterexample, but it has not been proven to be. However, I believe I might have found myself another potential counterexample, namely, $$387420547.$$ This number cannot be generated from any other numbers, as what can happen in Problem $5$. This is because in the problem, you can join primes together to make a new number that might be prime. In the number above, there are n prime numbers such that when you concatenate them, they make this number. After going to the link and reading the problem, you will understand why this is very important.
If you cannot access the link, please let me know, and I will write out the problem in this post for all to see. My question is, Is the number above an actual counterexample? It seems like it never climbs to a prime, no matter how many iterations I make. Can it be checked for certain?
Thank you in advance.