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| GoldenNumber.net |
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Powers of PhiPhi has a unique additive relationshipThe powers of phi have unusual properties in that they are related not only exponentially, but are additive as well. We know that: Phi 2 = Phi + 1 Which is the same as: Phi 2 = Phi 1 + Phi 0 And this leads to the fact that for any n: Phi n+2 = Phi n+1 + Phi n Thus each two successive powers of phi add to the next one!
Powers of Phi and its reciprocalAnother little curiosity involves taking phi to a power and then adding or subtracting its reciprocal: For any even integer n: Phi n + 1 / Phi n = a whole number For any odd integer n: Phi n - 1 / Phi n = a whole number Examples are shown in the tables below: for n = even integers
for n = odd integers
The whole numbers generated by this have a relationship among themselves, creating an additive series, similar in structure to the Fibonacci series, and which also converges on phi:
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