Multilevel Fibonacci Conversion and Addition

Recently we have been making studies [1, 2] on the Fibonacci number system that appears to be of considerable importance to the Fibonacci computer [3], A specific result that appears to be of particular interest is that through multilevel coefficients on a Fibonacci radix system, efficient extension of representations can occur. For example, through a ternary coefficient system, a doubling of the range with half the number of digits over a binary system has been shown, while in fact allowing a restriction to only even-subscripted Fibonacci numbers in the radix [1]. Consequently, further investigation of various other properties and extensions has seemed warranted. This has led us to our present studies which indicate that the Fibonacci computer may have added features, over those of redundancy for error detection and correction already reported [4]. One such added feature lies in efficient processing techniques with these being based upon the conversion of numbers into special forms, including evenand odd-subscripted Fibonacci radix systems. With this in mind, we develop in this paper several new Fibonacci number representations, in particular evenand odd-subscripted ones, a signed ternary one, and conversions between them. These ideas are developed in Sections II and III. In particular, we give the details of conversions among the various ternary Fibonacci radix representations. For reference purposes, we recall the defining recursion relation of the Fibonacci numbers [5] i = i-i + i-2> o = 0, F1 = 1. (l.D