Fibonacci and Lucas Numbers with Applications, Volume 1, 2nd Edition

COMPLETENESS THEOREMS

This chapter provides additional opportunities for studying patterns and making conjectures.

27.1 COMPLETENESS THEOREM

We begin with yet another interesting pattern:

Every integer on the LHS of each equation is a positive integer; and each number on the RHS is a Fibonacci number, and each occurs exactly once.

More generally, we have the following result.

Theorem 27.1 (Completeness Theorem)

Every positive integer can be expressed as the sum of a finite number of distinct Fibonacci numbers.

Proof

Let be the largest Fibonacci number . Then , where . Let be the largest number . Then , where . Continuing like this, we get , where . Since this sequence of decreasing ...