Given the inverse of the Golden Mean $ \tau^{ -1} = \phi = { 1\over 2} (\sqrt 5 - 1)$, it is known that the continuous fraction expansion of $ \phi^{ -1} = 1 + \phi = \tau$ is $ ( 1, 1, 1, \cdots )$. Integer solutions for the pairs of numbers $ ( d_i, n_i ), i = 1, 2, 3, \cdots $ are found obeying the equation $ ( 1 + \phi)^n = d + \phi^n$. The latter equation was inspired from El Naschie's formulation of fractal Cantorian space time $ {\cal E}_\infty$, and such that it furnishes the continuous fraction expansion of $ ( 1 + \phi )^n ~= ~ (d, d, d, d, \cdots )$, generalizing the original expression for the Golden mean. Hardy showed that is possible to demonstrate nonlocality without using Bell inequalities for two particles prepared in
nonmaximally entangled states. The maximal probability of obtaining his nonlocality proof was found to be precisely
ϕ5. Zheng showed that three-particle nonmaximally entangled states revealed quantum nonlocality without using inequalities, and the maximal probability of obtaining the nonlocality proof was found to be $ 0.25 \sim \phi^3 = 0.236$. Given that the two-parameter $ p, q$ quantum-calculus deformations of the integers $ [ n ]_{ p, q} = F_n $
coincide precisely with the Fibonacci numbers, as a result of Binet's formula when $ p = ( 1 + \phi) = \tau, q = - \phi = - \tau^{ -1}
,weexplorefurthertheimplicationsoftheseresultsinthequantumentanglementoftwo−particlespin−s$ states. We finalize with some remarks on the generalized Binet's formula corresponding to generalized Fibonacci sequences.