Zeckendorf representation of multiplicative inverses modulo a Fibonacci number

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