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Prove Phi's Peculiar Property (Posted on 2016-10-17) Difficulty: 3 of 5
The golden ratio number φ = (1+√5)/2 possesses many interesting properties.

inter alia
For any even integer n: φn + 1 /φn is an integer
For any odd integer n: φn - 1 /φn is an integer

Prove the above.

No Solution Yet Submitted by Ady TZIDON    
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A different approach (spoiler) | Comment 2 of 3 |

Outline solution

Using phi2 = phi + 1 repeatedly to break down phin to a

linear function of phi shows that   phin = Fn*phi + Fn-1 ,

where Fi are Fibonacci numbers (F1 = 1, F2 = 1, F3 = 2 …).

(for example, phi6 = F6*phi + F5 = 8*phi + 5)

This can be extended to negative values of n by using the

bidirectional Fibonacci series:  …5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5…

Note that for even n,  Fn = -F-n and for odd n, Fn = F-n .

Thus     phin + 1/phin      = (Fn*phi + Fn-1)+ (F-n*phi + F-n-1)

                                    = (Fn + F-n)*phi + Fn-1 + F-n-1      

so for even n the bracket is zero and the result is an integer.

Also       phin – 1/phin     = (Fn*phi + F-n) – (F-n*phi + F-n-1)

                                    = (Fn – F-n)*phi + F-n – F-n-1

so for odd n the bracket is zero and the result is an integer.

_________

PS Is there a simple way to produce greek letters in these

postings?  I’m pasting Word text.



  Posted by Harry on 2016-10-18 18:44:16
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