Powers of Phi

Gary Meisner

12 years ago

Phi has a unique additive relationship.

The 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!

n	Phi^ⁿ
0	1.000000
1	1.618034
2	2.618034
3	4.236068
4	6.854102
5	11.090170
6	17.944272

Here’s a little more phi mathemagic, contributed by Abe Ihmeari:

Φ * √5 = 3.6180339… = Φ + 2

Powers of Phi and its reciprocal:

Another 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

n	Phi^ⁿ	1/Phi^ⁿ	Phi^ⁿ + 1/Phi^ⁿ
0	1.000000000	1.000000000	2
2	2.618033989	0.381966011	3
4	6.854101966	0.145898034	7
6	17.944271910	0.055728090	18
8	46.978713764	0.021286236	47
10	122.991869381	0.008130619	123

for n = odd integers

n	Phi^ⁿ	1/ Phi^ⁿ	Phi^ⁿ – 1/Phi^ⁿ
1	1.618033989	0.618033989	1
3	4.236067977	0.236067977	4
5	11.090169944	0.090169944	11
7	29.034441854	0.034441854	29
9	76.013155617	0.013155617	76
11	199.005024999	0.005024999	199

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:

Exponent n	0	1	2	3	4	5	6	7	8	9	10	11
Result	2	1	3	4	7	11	18	29	47	76	123	199