# Relationship of powers of Phi to Lucas Numbers

I was watching a Numberphile and the interviewee was explaining various attributes of Lucas Numbers and he made the statement about creating a sequence by starting with the Golden Ratio and raising it to powers 1..n then rounding that number to generate a sequence of Lucas numbers.

I wondered what relationship (difference) the actual number had with the rounded number so I wrote some Python 3 code (I tried it in Octave also):

phi = (1 + math.sqrt(5)) / 2.0
for n in range(2000):
number = math.pow(phi, n)
print(n, round(number) - number)


It looks like after $\phi^{73}$ the value drops to zero (well, Python and Octave's 0.0) and never recovers on my machine. $\phi^{1259}$ produces an overflow. I just thought this was an interesting property and just was wondering:

1. Are the power values after 73 really zero or just too small for my 64-bit machine?
2. Is the sequence shown here used or applied anywhere? I searched in some of the Lucas literature but nothing popped out about the differences.
3. Does anyone know if there is any more variance after 73?

• It's just double precision arithmetic. Generally, you have $L_n = \phi^n + \psi^n$, where $\psi = 1-\phi = \frac{1-\sqrt{5}}{2}$. – Daniel Fischer Oct 6 '14 at 12:33