2020/10/24 by Hung Viet Chu, Chu, Hung Viet
Mathematics · #11B39 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11B39
paper · pdf · doi:10.48550/arxiv.2010.15592
6 pages
arxiv created 2020/10/24 · arxiv updated 2020/10/30
Zeckendorf proved that every positive integer has a unique partition as a sum of non-consecutive Fibonacci numbers. We study the difference between the number of summands in the partition of two consecutive integers. In particular, let L(n) be the number of summands in the partition of n. We characterize all positive integers such that L(n) > L(n+1), L(n) < L(n+1), and L(n) = L(n+1). Furthermore, we call n+1 a peak of L if L(n) < L(n+1) > L(n+2) and a divot of L if L(n) > L(n+1) < L(n+2). We characterize all such peaks and divots of L.