Problem 2

Prove that every positive integer can be written as a finite sum of distinct integral powers of the golden mean

\phi=\frac{1+\sqrt5}{2}

. Here an integral power is

\phi^i

with

i

an integer, not necessarily positive.

Step 3 of 5: Finish when the unit digit is zero

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a_0=0\Longrightarrow N=(N-1)+\phi^0

Detailed analysis

If the normalized expansion of $N-1$ has $a_0=0$ , then adding $\phi^0=1$ gives $N$ as a sum of distinct integral powers: exponent zero was absent, so no term is repeated.