On Quaternion Gaussian Bronze Fibonacci Numbers

On the Fibonacci Numbers

The Fibonacci numbers are terms of the sequence defined in a quite simple recursive fashion. However, despite its simplicity, they have some curious properties which are worth attention. In this set of notes, we will look at some of the important features of these numbers. In the first half of the notes, our attention shall be paid to the relationship of the Fibonacci numbers and the Euclidean ...

Some Operations on Quaternion Numbers

(1) <(z1 · z2) = <(z2 · z1). (2) If z is a real number, then z + z3 = <(z) + <(z3) + =1(z3) · i+ =2(z3) · j + =3(z3) · k. (3) If z is a real number, then z − z3 = 〈<(z)−<(z3),−=1(z3),−=2(z3), −=3(z3)〉H. (4) If z is a real number, then z · z3 = 〈<(z) · <(z3),<(z) · =1(z3),<(z) · =2(z3),<(z) · =3(z3)〉H. (5) If z is a real number, then z · i = 〈0,<(z), 0, 0〉H. (6) If z is a real number, then z · j...

Some Operations on Quaternion Numbers

In this article, we give some equality and basic theorems about quaternion numbers, and some special operations. the notation and terminology for this paper. In this paper z 1 , z 2 , z 3 , z 4 , z are quaternion numbers. The following propositions are true: (1) (z 1 · z 2) = (z 2 · z 1). (2) If z is a real number, then z + z 3 = (z) + (z 3) + 1 (z 3) · i + 2 (z 3) · j + 3 (z 3) · k. (4) If z i...

ON r-GENERALIZED FIBONACCI NUMBERS

Fibonacci Numbers

One can prove the following three propositions: (1) For all natural numbers m, n holds gcd(m,n) = gcd(m, n + m). (2) For all natural numbers k, m, n such that gcd(k, m) = 1 holds gcd(k,m · n) = gcd(k, n). (3) For every real number s such that s > 0 there exists a natural number n such that n > 0 and 0 < 1 n and 1 n ¬ s. In this article we present several logical schemes. The scheme Fib Ind conc...