Zeckendorf representation of multiplicative inverses modulo a Fibonacci number

Prempreesuk, Noppakaew, and Pongsriiam determined the Zeckendorf representation of the multiplicative inverse of 2 modulo $$F_n$$ F n , for every positive integer n not divisible by 3, where $$F_n$$ F n denotes the nth Fibonacci number. We determine the Zeckendorf representation of the multiplicative inverse of a modulo $$F_n$$ F n , for every fixed integer $$a \ge 3$$ a ≥ 3 and for all positive integers n with $$\gcd (a, F_n) = 1$$ gcd ( a , F n ) = 1 . Our proof makes use of the so-called base-$$\varphi $$ φ expansion of real numbers.