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

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.

$$\frac{1}{\alpha^b}\sum_{i=1}^{10} \alpha^i= \frac{(\alpha^1 + \alpha^2 + \alpha^3 + \alpha^4 + \alpha^5 + \alpha^6 + \alpha^7 + \alpha^8 + \alpha^9 + \alpha^{10})}{\alpha^b} = c$$