Eser, Engin; Goy, Taras; Özkan, Engin

Mersenne numbers, introduced by Marin Mersenne in the early 17th century and known since antiquity for their deep links to prime number theory, have long fascinated mathematicians. In this study, we introduce a second-order sequence derived from consecutive Mersenne numbers, which we refer to as the Trisenne sequence. Our objective is to investigate its structural properties and to establish relationships between the Trisenne numbers and the general Horadam (or generalized Fibonacci) sequence through their generating functions. The connection between these families is analyzed using both ordinary and exponential generating functions, with particular emphasis on classical cases such as the Fibonacci, Lucas, Pell, and Jacobsthal sequences. We also discuss the historical context and mathematical lineage of Horadam sequences, tracing their origin to the work of Edouard Lucas and Alwyn Horadam. Theoretical results are illustrated with explicit examples, and several related identities are presented.

Alkan, Aslı; Kayalar, Mehmet; Bulut, Hasan

In this paper, the modified Kudryashov method is utilized to construct the exact traveling solutions to the Hirota-Ramani equation. The Hirota-Ramani equation holds significant importance as a fundamental model in the examination of nonlinear and integrable systems. It offers valuable theoretical insights and practical applications across multiple domains of physics and applied mathematics. The modified Kudryashov method was utilized to acquire the novel solutions of the Hirota-Ramani equation. Consequently, numerous analytical exact solutions have been derived, including rational, trigonometric, and hyperbolic function solutions. This method is potent, effective, and serves as an option for developing new solutions to many sorts of fractional differential equations utilized in mathematical physics.

Baskonus, Haci Mehmet; Raihen, Md Nurul; Kayalar, Mehmet

In this paper, we apply the powerful sine-Gordon expansion method (SGEM), along with a computational program, to construct some new traveling wave soliton solutions for two models, including the higher-order nonlinear Boussinesq dynamical wave equation, which is a well-known nonlinear evolution model in mathematical physics, and the (1+1)-dimensional framework of the Van der Waals gas system. This study presents some new complex traveling wave solutions, as well as logarithmic and complex function properties. The 3D and 2D graphical representations of all obtained solutions, unveiling new properties of the considered model are simulated. Additionally, several simulations, including contour surfaces of the results, are performed, and we discuss their physical implications. A comprehensive conclusion is provided at the end of this paper.

Kızıltuğ, Sezai; Erişir, Tülay; Mumcu, Gökhan; Yaylı, Yusuf

In this study, the curve theory, which occupies a very important and wide place in differential geometry, has been studied. One of the most important known methods used to analyze a curve in differential geometry is the Frenet frame, which is a moving frame that provides a coordinate system at each point of the curve. However, the Frenet frame of any curve cannot be constructed at some points. In such cases, it is useful to define an alternative frame. In this study, instead of the Frenet frame that characterizes a regular curve in Euclidean space E3, we have defined a different and new frame on the curve. Since this new frame is defined with the aid of the Darboux vector, it is very compatible compared to many alternative frames in application. Therefore, we have named this new frame the “modified adapted frame” denoted by {N∗, C∗, W∗ }. Then, we have given some characterizations of this new frame. In addition to that, we have defined N∗-slant helices and C∗-slant helices according to {N∗, C∗, W∗ }. Moreover, we have studied C∗-partner curves using this modified adapted frame. Consequently, by investigating applications, we have established the relationship between C∗-partner curves and helices, slant helices.

Erişir, Tülay

Spinors permeate all of modern physics and have also an important place in mathematics. Spinors are used intensively in modern theoretical physics and differential geometry. In this study, spinors are used for a different representation of differential geometry in E3. The goal of this study is also the spinor structure lying at the basis of differential geometry. In this paper, firstly, spinors are introduced algebraically. Then, the spinor construction of Bertrand curves is defined. Moreover, the angle notion for these spinors is given. In this way, a different geometric construction of spinors is established in this paper.