Way back, on March 14, 2003, I was a true nerd. Standing in line for the sneak preview of The Matrix Reloaded, I had a Ti-83 Plus graphing calculator in hand
and was playing around with the Fibonacci sequence when I noticed that if you take the logarithm base φ of the Fibonacci sequence the result was...interesting.
From there I discovered a inverse Fibonacci formula that I was unable to find anywhere on the internet:
I worked with several people in my High School and nearby Appalachain State University for assistance working on a publishable proof. Dr. William Bauldry
mentored me to develop a proof and I had hoped to publish the finding in a journal. Unfortunately, publication never came, and now, seven years later, I stumbled
across some of the old paperwork and I decided to make it public property.
Here's a PDF version of the old paper. This might not even be the last version of the paper I worked on, but here it is: fibonacci.pdf
Also in 2003, I worked with a good friend Malcom Mollison to research Fibonacci Primes. The research incorporated the above theorem. The research may have been lost...ah well the general idea is easy enough to implement. Our approach was largely computational (compute primes, count their index, determine if they are Fibonacci numbers, if so compute Fibonacci index and output all four values). This approach led to some interesting assumptions, but did not definively prove anything and probably never would; computational approaches to finding truth in primes are historically fruitless.
If you're reading the paper, please remember I wrote this when I was a hotshot 17-year-old who thought he was smart. I've made no attempts to cleanup the paper, just thought I'd share it with anyone interested in the subject.