Now that James Davis has found a counter example, 13532385396179, to John Conway's climbtoaprime conjecture, I would be interested to learn whether this has any implications of interest in number theory.

2$\begingroup$ I guess the answer would be no, but demonstrating that might be impossible. Are there any implications of interest in various other puzzles/results involving patterns in base10 digits of a number? $\endgroup$ – spin Jun 9 '17 at 15:41

2$\begingroup$ Do there exist more nonprime fixed points than a heuristic argument would suggest? If so then that might lead to an interesting mathematical insight. $\endgroup$ – Timothy Chow Jun 9 '17 at 20:28

2$\begingroup$ 100 years ago, Dudeney noticed $2^59^2=2592$. If only 9 were prime.... $\endgroup$ – Gerry Myerson Jun 10 '17 at 5:53

$\begingroup$ It would be helpful to add the link (cheswick.com/ches/conway1000.pdf) to the list of Conway's problems containing his "climbtoaprime conjecture" under number 5. $\endgroup$ – Taras Banakh Feb 8 '18 at 7:01
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!

7$\begingroup$ Always cool when the mathematician mentioned gives his two cents! If only we could get Conway on here... $\endgroup$ – user78249 Jun 9 '17 at 22:49