Electronic Theses and Dissertation

Penelitian ini membahas sifat-sifat keterbagian pada bilangan Fibonacci, sebuah barisan bilangan yang didefinisikan secara rekursif dan memiliki keterkaitan luas dengan teori bilangan, aljabar, serta kombinatorika. Penelitian ini mengkaji sejumlah dugaan keterbagian yang ditemukan secara empiris, mencakup keterkaitan kelipatan, pembagi persekutuan terbesar, dan keprimaan bilangan Fibonacci. Hasil penelitian menunjukkan bahwa: (1) dua bilangan Fibonacci berurutan selalu relatif prima, (2) Jika m | n maka F_m | F_n, dan sebaliknya, (3) untuk m,n ≥1 berlaku gcd⁡(F_m,F_n )=F_gcd⁡(m,n), (4) Sifat m=qn+r yang berakibat gcd⁡(m,n)=gcd⁡(n,r) juga diikuti oleh bilangan Fibonacci, yaitu gcd⁡(F_m,F_n )=gcd⁡(F_n,F_r ), serta (5) syarat perlu agar F_n prima adalah n sendiri harus prima. Selain itu, penelitian ini juga membahas beberapa sifat populer bilangan Fibonacci, identitas yang berkaitan dengan jumlah suku, teka-teki paradoks luas berbasis Fibonacci, serta pendekatan eksplisit untuk memperkirakan banyaknya digit bilangan Fibonacci.
Kata Kunci: Bilangan Fibonacci, keterbagian, gcd, periode pisano.

This research discusses some divisibility properties of Fibonacci numbers, a sequence of numbers defined recursively and broadly connected to number theory, algebra, and combinatorics. The study examines several empirically observed conjectures on divisibility, including relationships involving divisibility, the greatest common divisor, and the primality of Fibonacci numbers. The results shows that: (1) two consecutive Fibonacci numbers are always relatively prime, (2) if m | n, then F_m | F_n, and vice versa, (3) for m,n ≥1, the identity gcd⁡(F_m,F_n )=F_gcd⁡(m,n) holds, (4) the property m=qn+r, which implies gcd⁡(m,n)=gcd⁡(n,r), is also satisfied by Fibonacci numbers, namely that gcd⁡(F_m,F_n )=gcd⁡(F_n,F_r ), and (5) a necessary condition for F_n to be prime is that n itself is must be a prime. in addition, this research also discusses several well-known properties of Fibonacci numbers, identities related to sums of terms, a Fibonacci-based area paradox, and an explicit approach for estimating the number of digits of Fibonacci numbers. Keywords: Fibonacci numbers, divisibillity, gcd, prime numbers, Pisano period.