Zeckendorf representation of multiplicative inverses modulo a Fibonacci number
Gessica Alecci, Nadir Murru, Carlo Sanna · Monatshefte für Mathematik · 2022
Abstract Prempreesuk, Noppakaew, and Pongsriiam determined the Zeckendorf representation of the multiplicative inverse of 2 modulo $$F_n$$ Fn , for every positive integernnot divisible by 3, where $$F_n$$ Fn denotes thenth Fibonacci number. We determine the Zeckendorf representation of the multiplicative inverse ofamodulo $$F_n$$ Fn , for every fixed integer $$a \ge 3$$ a≥3 and for all positive integersnwith $$\gcd (a, F_n) = 1$$ gcd(a,Fn)=1 . Our proof makes use of the so-called base- $$\varphi $$ φ expansion of real numbers.