# Not your standard base

**Number Theory**Level 3

A number in the *Fibonacci* base is defined as follows:

The number is composed of a string of 1's and 0's (like binary)

No two consecutive 1's occur in any number (e.g. \(101000101_{F} \) is a number in the Fibonacci base.)

The place value of each digit from left to right is corresponding to the Fibonacci sequence

Conversion to base 10 requires multiplying the place value by it's corresponding digit. e.g.

\[\begin{align} 100101_{F} &= 1 \times F_1 + 0 \times F_2 + 1 \times F_3 + 0 \times F_4 + 0 \times F_5 + 1 \times F_6\\ &= 1 + 3 + 13\\&=16 \end{align}\]

Do all natural numbers have a representation in the Fibonacci base?

The Fibonacci sequence is defined as:

\[F_1 = 1, F_2 = 2, F_n = F_{n-1} + F_{n-2} \quad \text{for } n > 2\]