Seminario: "Zeckendorf representation of multiplicative inverses modulo a Fibonacci number" - Gessica Alecci
"Zeckendorf representation of multiplicative inverses modulo a Fibonacci number"
Let (F_n) be the sequence of Fibonacci numbers, which is defined by the initial conditions F1 = F2 = 1 and by the linear recurrence F_n = F_{n−1} + F_{n−2} for n ≥ 3. Every positive integer n can be written as a sum of distinct non-consecutive Fibonacci numbers, that is, n = sum_{i=1}^m d_i F_i , where m ∈ N, d_i ∈ {0, 1}, and d_i d_{i+1} = 0 for all i ∈ {1, . . . , m − 1}. This is called the Zeckendorf representation of n and, apart from the equivalent use of F_1 instead of F_2 or vice versa, is unique. The Zeckendorf representation of integer sequences has been studied in several works. For instance, Filipponi and Freitag studied the Zeckendorf representation of numbers of the form F_{kn} / F_n , F_n^2 / d and L_n^2 / d, where L_n are the Lucas numbers and d is a Lucas or Fibonacci number. Filipponi, Hart, and Sanchis analyzed the Zeckendorf representation of numbers of the form mF_n. Filipponi determined the Zeckendorf representation of mF_n F_{n+k} and mL_n L_{n+k} for m ∈ {1, 2, 3, 4}. Prempreesuk, Noppakaew, and Pongsriiam determined the Zeckendorf representation of the multiplicative inverse of 2 modulo F_n, for every positive integer n not divisible by 3, where Fn denotes the nth Fibonacci number. We determine the Zeckendorf representation of the multiplicative inverse of a modulo F_n, for every fixed integer a ≥ 3 and for all positive integers n with gcd(a, F_n ) = 1. Our proof makes use of the so-called base-φ expansion of real numbers.
Coauthors: Nadir Murru, Carlo Sanna