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Jun 13, 2011 at 16:59 comment added Eric Naslund @Will Jagy: I added a proof of your observation at the end of my answer.
Jun 13, 2011 at 11:18 comment added Emil Jeřábek @Eric: $F_{12}=12^2$ is divisible by $3$.
Jun 13, 2011 at 2:53 comment added Eric Naslund Also, you might have used this, but I'll put the link for any others reading the comments who want to play around with numerics: This page: mersennus.net/fibonacci has lists of the complete factorization of the first 1000 Fibonacci numbers and the first 1000 Lucas Numbers.
Jun 13, 2011 at 2:05 history answered Will Jagy CC BY-SA 3.0