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 5 of 5: Complete the induction

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

Detailed analysis

The carry procedure preserves the value and produces a normalized expansion of $N-1$ with $a_0=0$ . Adding the distinct term $\phi^0$ proves the assertion for $N$ ; the base case is $1=\phi^0$ .