Powers of Phi

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:

Φ² = Φ + 1

Which is the same as:

Φ² = Φ¹ + Φ⁰

And this leads to the fact that for any n:

Φⁿ⁺² = Φⁿ⁺¹ + Φⁿ

Thus each two successive powers of phi add to the next one!

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:

Φⁿ  +  1/Φⁿ = a whole number

For any odd integer n:

Φⁿ  -  1/Φⁿ = 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: