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Fibonacci Factors

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?

Plus or Minus

Age 16 to 18 Challenge Level:

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.
This problem complements the material in the article The Golden Ratio, Fibonacci Numbers and Continued Fractions.

For a sequence of, mainly more elementary, problems on these topics see Golden Mathematics.