The Golden Number hype is mainly humbug, but its core does contain some mathematical facts. It is a long standing tradition to blow these up to pompous mysteries. The founding father of this dubious practice is Luca Pacioli, who in his Divine [sic] Proportion (1509) gives all kinds of overblown, even theological, epithets to mathematical properties known since Euclid. Today is hardly different, and one may find simple mathematical facts described at length and analysed in depth as if the golden number were unique in having those properties. Most people seem unaware of the fact that the golden number is but one of a kind, and that there is nothing very special about preserving its decimal part under inversion (wow, 1/0.61803...=1.61803...), or being the limit of a simple sequence of fractions (wow, approximately a quotient of Fibonacci numbers).
This episode of Golden Maths is devoted to Elementary arithmetic. Actually, by 'arithmetic' we mean 'non-geometry'; limits, for instance, are included. And 'elementary' implies there is less elementary to follow. For now, we consider the golden number as senior member of the family of metal numbers. In recent times there is some tendency to extend the golden epithet (of 19th century German origin, as we know) to silver, bronze and generally metal(lic). It's funny, and therefore recommendable in view of the deadly (I would say leaden) seriousness of the golden prophets.
So here is the victory stand with the gold, silver and bronze finalists in the grand metal race:
There are infinitely many more competitors (all staying in Hilbert's hotel, BTW 😄), and they look very much alike. All you will ever need to know about them is on the blackboard below. For gold, silver, bronze plug in T=1,2,3.
The (elementary) proofs are in this single sheet:
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