The Zeckendorf representation of a positive integer is the unique expression of this integ...

The Zeckendorf representation of a positive integer is the unique expression of this integer as the sum of distinct Fibonacci numbers, where no two of these Fibonacci numbers are consecutive terms in the Fibonacci sequence and where the term f1 = 1 is not used (but the term f2 = 1 may be used).

What is wrong with the claim that an 8 × 8 square can be broken into pieces that can be reassembled to form a 5 × 13 rectangle as shown?

(Hint: Look at the identity in Exercise. Where is the extra square unit?)

Exercise

Prove that for every positive integer n.