Henry, my previous post had two references to base phi that are both from the Mathematics Magazine, so pretty accessible. I can send you pdfs if you want. Representation as sums of Fibonacci numbers aren’t strictly a “base” so I didn’t mention them earlier, but they have a long history, and are often called “Zeckendorf notation” after their discoverer. A nice paper that talks about various difference equations and as well as putting continued fractions in the same setting is A.S. Fraenkel, Systems of Numeration, Am. Math. Monthly 92(2) 1985. Blowing my own trumpet, a recent paper of mine presented at the first MOVES conference shows how Zeckendorf representation as a way of representing variable precision integers is particularly good, and is stunningly useful for representing continued fraction partial quotients. I also pull together references on arithmetic in Zeckendorf representation and add a few wrinkles of my own. And it is just enough different from base phi representation to be interesting. A copy can be found at http://educ.jmu.edu/~lucassk/Papers/MOVES%20paper%20revised.pdf There are some lovely relationships between Zeckendorf notation and base phi, but one of the more elegant uses Lucas numbers (no relation!). You can represent numbers as sums of Lucas numbers just like Fibonacci. Then Lucas numbers satisfy L_n= phi^n + (-phi)^n, and you can manipulate this into base phi after dealing with the negative for odd n. Steve -- On May 16, 2018, at 10:37 AM, Henry Baker <hbaker1@pipeline.com<mailto:hbaker1@pipeline.com>> wrote: The Fibonacci number representation stems from the *difference equation* for Fibonacci numbers, rather than from a traditional positional number system based on phi or 1/phi (or -phi or -1/phi). In my Google Scholar literature search, I found a number of papers investigating different number systems based upon *difference equations* which were inspired by Fibonacci, but essentially nothing about positional number systems based upon phi.