@MartinHopf A curious semiprime with our current year is $F_{2024}-2024$. I currently try to factor $F_{2024}+2024$. $F_n-n$ and $F_n+n$ are both prime for $n=8 , 28 , 808$ (All with ending digit $8$. What is the next such $n$ ?) I also currently try to solve the puzzle.

My current best number for the puzzle is $$19384156968019092965661281715951$$

@Sahaj I found a second solution for $n=6$ , but none for the other $n\le 10$

Hmm. That is interesting.

for $n=2$, I couldn’t prove it. All I know is that $a=2x$, $b=2y$, $c=2z$, where x,y,z are all odd and with gcd 1.