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In most sources, the Golden Ratio has been referred to as: $\phi = \frac{\sqrt{5}+1}{2}$. Nevertheless, in the book " The Theory of Numbers, a Text and Source Book of Problems ", by Prof. Andrew Adler and Prof. John E. Coury, it has been alluded to differently.

... For example, any two successive terms have no divisor in common except 1, and the ratio of sufficiently large successive terms is arbitrarily close to the "golden ratio" ($\frac{\sqrt{5}-1}{2}$), which was of interest to the ancient Greeks ...

- The Theory of Numbers, a Text and Source Book of Problems

I wish to know whether the both of them are correct or not? If yes, why?

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The Golden Ratio mentioned in the book is the reciprocal of the commonly used Golden Ratio $$ \frac{\sqrt5-1}2\frac{\sqrt5+1}2=\frac{5-1}4=1 $$

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