On General Divisibility of Sums of Integral Powers of the Golden Ratio

Herta T. Freitag and Daniel C. Fielder

A chapter in Applications of Fibonacci Numbers, 1999, pp 149-154 from Springer

Abstract: Abstract The well-known Golden Ratio, $$\alpha = (1 + \sqrt {5} )/2 $$ , is the limit as n→∞ of the ratio of the Fibonacci numbers F n/F n-1 and the Lucas numbers L n/L n-1. Eq. (1) served to illustrate in [2,3] that unique integer solutions b = 7 and c = 11 exist. $$\frac{1}{\alpha^b}\sum_{i=1}^{10} \alpha^i= \frac{(\alpha^1 + \alpha^2 + \alpha^3 + \alpha^4 + \alpha^5 + \alpha^6 + \alpha^7 + \alpha^8 + \alpha^9 + \alpha^{10})}{\alpha^b} = c$$

Keywords: 11B39; 05A15 (search for similar items in EconPapers)
Date: 1999
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DOI: 10.1007/978-94-011-4271-7_15

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