Open Journal of Mathematics and Physics | Volume 1, Article 63, 2019 | ISSN: 2674-5747 https://doi.org/10.31219/osf.io/3tqeg | published: 11 Dec 2019 | https://ojmp.wordpress.com CK
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Greatest Common Divisors of the Fibonacci numbers and their indices
Open Mathematics Collaboration∗† December 30, 2019
Abstract A very interesting property of the Fibonacci numbers alongside the steps involved in its proof are presented.
keywords: greatest common divisor, gcd, Fibonacci numbers
Notation
1. a, b, i, j, m, n, q, r ∈ N = {1, 2, 3, ...}
2. Let fi be the Fibonacci numbers. 3. (i, j) is the greatest common divisor (gcd) of i and j.
4. iSj means i divides j.
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1 Theorem
5.
(fi, fj) = f(i,j)
Steps required to prove the theorem
6. The following steps prove (5) [1].
7. Suppose i ≥ j.
8. Use the Euclidean division, i = jq + r.
9. Lemma 1: fi+j = fifj+1 + fi−1fj.
10. Lemma 2: iSj → fiSfj.
11. Lemma 3:
∃(a, b − na) → (∃(a, b) ∧ ((a, b) = (a, b − na))).
12. Lemma 4: Two consecutive Fibonacci numbers are coprimes.
13. Lemma 5: ((a, b) = 1) → ((ac, b) = (c, b)).
14. Lemma 6: (((a, b) = 1) ∧ (aSc) ∧ (bSc)) → (abSc).
15. Lemma 7. Let: a0, a1, a2, ... be a sequence of natural numbers; a0 = 0; ∀m ≥ n, (am, an) = (an, ar); r is the rest of the division of m by n. Then (am, an) = a(m,n).
2 Final Remarks
16. The calculations outlined in the previous section are left as an exercise.
17. Theorem (5) states that the greatest common divisor (gcd) of two Fibonacci numbers is equal to the (gcd)-th Fibonacci number.
Open Invitation
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Ethical conduct of research
This original work was pre-registered under the OSF Preprints [2], please cite it accordingly [3]. This will ensure that researches are conducted with integrity and intellectual honesty at all times and by all means.
References
[1] Hefez, Abramo. “Aritmética.” Rio de Janeiro: SBM (2016).
[2] COS. Open Science Framework. https://osf.io
[3] Lobo, Matheus P. “Greatest Common Divisors of the Fibonacci Numbers and Their Indices.” OSF Preprints, 11 Dec. 2019. https://doi.org/10.31219/osf.io/3tqeg
The Open Mathematics Collaboration
Matheus Pereira Lobo (lead author, [email protected])1,2
3 1Federal University of Tocantins (Brazil); 2Universidade Aberta (UAb, Portugal)
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