The golden ratio as a number base

The Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …) are one of the most famous sequences of integers. Every number in the sequence after the first two is the sum of the two previous numbers: 1+1=2, 1+2=3, 2+3=5, and so on. It turns out that every positive integer can be written as a sum of distinct Fibonacci numbers, possibly in several differen…