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arXiv · 2203.06101

Zeckendorf representation of multiplicative inverses modulo a Fibonacci number

Gessica Alecci·Nadir Murru·Carlo Sanna
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Abstract

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 $F_n$ denotes the $n$th Fibonacci number. We determine the Zeckendorf representation of the multiplicative inverse of $a$ modulo $F_n$, for every fixed integer $a \geq 3$ 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.

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Gessica Alecci, Nadir Murru, Carlo Sanna. 2022-03-11. Zeckendorf representation of multiplicative inverses modulo a Fibonacci number. https://arxiv.org/abs/2203.06101

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