I noticed in my mathematical meanderings involving Phi I often had to reduce algebraic formulae involving powers of Phi. I began to wonder if there was an elegant way to represent those powers that is both useful and simple. I first began with the positive integer powers, 1, 2, 3, ….
f
0 = 1f
1 = ½[Ö 5 + 1]f
2 = ¼[2Ö 5 + 6] = ½[Ö 5 + 3]f
3 = ¼[4Ö 5 + 8] = ½[2Ö 5 + 4]f
4 = ½[3Ö 5 + 7]f
5 = ¼[10Ö 5 + 22] = ½[5Ö 5 + 11]f
6 = ¼[16Ö 5 + 36] = ½[8Ö 5 + 18]f
7 = ½[13Ö 5 + 29]f
8 = ¼[42Ö 5 + 94] = ½[21Ö 5 + 47]f
9 = ¼[68Ö 5 + 152] = ½[34Ö 5 + 76]And so I noticed the Fibonacci series as the first coefficient of the radical easily enough, and I speculated about the second term for a short bit, quickly verifying that it is indeed consecutive elements in the Lucas series. Thus, a nice way to represent the general form for all non-negative integer powers of Phi yields:
f
n = ½[F(n)Ö 5 + L(n)], n ³ 0 (1.1)Nice! Further analysis of the negative powers followed thus:
f
-1 = f - 1 = ½[Ö 5 - 1]f
-2 = ½[3 - Ö 5]f
-3 = ¼[4Ö 5 - 8] = ½[2Ö 5 - 4]f
-4 = ½[7 - 3Ö 5]f
-5 = ½[5Ö 5 - 11]The pattern here was quickly becoming apparent, resulting in the following for all non-positive integers n:
f
n = ½[(-1)n – 1F(|n|)Ö 5 + (-1)n L(|n|)], n £ 0 (1.2)Combining the two resulted in the following relation over all integers n:
f
n = ½[(-1)(a)F(|n|)Ö 5 + (-1)(b) L(|n|)] (1.3)where:
|
n |
even and negative |
odd and negative |
0 |
even and positive |
odd and positive |
|
a |
1 |
0 |
0 |
0 |
0 |
|
b |
0 |
1 |
0 |
0 |
0 |
Or, a = n – 1, b = n when n is negative, a = b = 0 all other n.
In all of the above formulae F(n) represents the nth member of the Fibonacci series, where F(0) = 0, F(1) = 1, and F(n+2) = F(n) + F(n + 1). Likewise L(n) denotes the nth member of the Lucas series, where L(0) = 2, L(1) = 1, and L(n+2) = L(n) + L(n + 1).
Some interesting twists on the above give the following Fibonacci and Lucas relationships:
F(n)
= (2fn - L(n))/Ö 5 (1.4)and
L(n)
= 2fn - F(n)Ö 5 (1.5)I have not seen these five relations published, either on the Internet or on paper, having read a dozen or so books on the subject. I have not gotten hold of S. Vajda’s "Fibonacci and Lucas Numbers, and the Golden Section", Ellis Horwood Limited, Chichester, 1989. I have a feeling they may be printed there, but until such time as I can verify, as the publication is currently out of print, I hope these will stand as my contributions.