Now that James Davis has found a counter example, 13532385396179, to John Conway's climb-to-a-prime conjecture, I would be interested to learn whether this has any implications of interest in number theory.
Hans said that Conway's point in asking it was that there exist problems easy to state but impossible to prove. The point I took away was that there exist problems that look so hard, nobody has tried anything easy.
Read the letter to Conway on Numberphile if you want to see how it was easy. I'll ask Conway if it has any other implications (if I get the chance). I kinda doubt it, but who knows. The idea isn't really limited to base 10. The problem and short search that worked generalizes to other bases. I worked in smaller bases at first.
Edit: If anyone does think of a mathematical usefulness to this, please let me know!