I've been looking at this problem trying to figure it out for awhile, but haven't been able to come up with a distinct proof of this: Do you think it's possible to express every positive integer as the sum of non-consecutive Fibonacci numbers? For example, 20 = 13 + 5 + 2, 33 = 21 + 8 + 3 + 1, and 34 = 34. I worked through some of this and came to the conclusion that for some numbers, the Fibonacci number directly below the chosen positive integer will always be used in the sum. [itex] 33 = F_8 + F_6 + F_4 + F_2 = 21 + 8 + 3 + 1. [/itex] Any suggestions?