In this paper, we define the 2k-step Jordan-Fibonacci sequence, and then we study the 2k-step Jordan-Fibonacci sequence modulo m. Also, we obtain the cyclic groups from the multiplicative orders of the generating matrix of the 2k-step Jordan-Fibonacci sequence when read modulo m, and we give the relationships among the orders of the cyclic groups obtained and the periods of the 2k-step Jordan-Fibonacci sequence modulo m. Furthermore, we extend the 2k-step Jordan-Fibonacci sequence to groups, and then we examine this sequence in the finite groups. Finally, we obtain the period of the 2k-step Jordan-Fibonacci sequence in the generalized quaternion group Q 2 n as applications of the results produced.MSC: Primary 11B50; 20F05; secondary 11C20; 20D60

The key agreement scheme is an important part of the cryptography theory. The first study in this field belongs to Diffie–Hellman and Merkle. We present a new key agreement scheme using a group action of special orthogonal group of 2 × 2 matrices with real entries on the complex projective line.

In this study, we define hyperbolic-type k-Fibonacci numbers and then give the relationships between the k-step Fibonacci numbers and the hyperbolic-type k-Fibonacci numbers. In addition, we study the hyperbolic-type k-Fibonacci sequence modulo m and then we give periods of the Hperbolic-type k-Fibonacci sequences for any k and m which are related the periods of the k-step Fibonacci sequences modulo m. Furthermore, we extend the hyperbolic-type k-Fibonacci sequences to groups. Finally, we obtain the periods of the hyperbolic-type 2-Fibonacci sequences in the dihedral group D2m, (m ? 2) with respect to the generating pairs (x,y) and (y, x).

In this manuscript, a new family of ݇ − Gaussian Fibonacci numbers has been identified and some relationships between this family and known Gaussian Fibonacci numbers have been found. Also, I the generating functions of this family for ݇ = 2 has been obtained.

No abstract