Ahmet Daşdemir

We determine all integer triples ( k, m, n ) satisfying the Diophantine equations P _k = J _m J _n and J _k = P _m P _n , thereby characterizing Pell numbers that are products of two Jacobsthal numbers, and vice versa. In addition, we study the intersection of the Pell and Jacobsthal sequences and show that the only common values occur when m = 1, corresponding to the integers 1 and 5.

This paper establishes fundamental algebraic identities for Stakhov hyperbolic functions, a recent generalization of hyperbolic functions based on recurrence sequences with functional parameters. We derive Vajda-type additivity relations, d’Ocagne’s formulas, Catalan and Cassini-type multiplicative laws, Gelin–Cesàro–type identities, and generating functions through Binet’s formulas, and introduce a novel platinum matrix framework. The matrix methodology thus generates further identities—including Honsberger-type decompositions and shift formulas—thereby unifying discrete recurrences with continuous symmetries. Special cases recover classical identities for hyperbolic Fibonacci functions, and new results emerge for Pell, Jacobsthal, and Fermat-type generalizations. The unified framework bridges recurrence sequences and hyperbolic function theory and demonstrates applications in differential geometry.

This study aims to find all possible solutions (k,m,n) of the Diophantine equations P_k=R_m R_n and R_k=P_m P_n . Our proof is conducted with the famous Matveev theorem and Dujella-Pethő reduction lemma.

Presented herein is a novel approach to apply the finite element method (FEM) to the forced vibration analysis of a two-axially preloaded plate with two constituent materials standing on a rigid foundation under a time-variable angled force based on a piecewise homogeneous body model. The article is pioneering the application of the FEM to analyze the mentioned problem, marking a significant advancement in the understanding of multilayered composite structures in engineering dynamics. First, the geometry of the problem and the physical considerations are explained in detail. Next, a mathematical solution process is developed in the context of the theory of linearized waves in elastic solids under a preloaded state according to the variational calculus. The solution algorithm is validated by presenting an error analysis thanks to norm functions and the convergence analysis compared to the available studies. Next, a thorough numerical discussion is performed to highlight how various factors involving the initial deformation state, the angles of inclined force, and the resonance frequency, together with the thickness ratio and the frequency response of the system, affect its forced vibration behaviors. Numerical results demonstrate that while increasing the vertical length of the body exceeds the resonance frequency of the system, increasing the horizontal lengths promotes that mode. Also, it is shown that when the Young modulus of the lower layer is bigger than that of the upper layer, the stability of the system increases.

This study aims to explore all Pell numbers that are the product of two random Pell-Lucas numbers and all Pell-Lucas numbers that are the product of two random Pell numbers based on linear forms in logarithms of algebraic numbers using Matveev's theorem and Dujella - Pethő reduction lemma. Further, we find all the common terms of Pell and Pell-Lucas numbers and show that no Pell and no Pell-Lucas numbers can be written as a square of another.

This paper presents an investigation into the generalization of hyperbolic Fibonacci sine and cosine functions, as well as Fibonacci spirals. Initially, we establish the main definitions and theoretically model them, listing several special cases. We then uncover fundamental results, including the De Moivre and Pythagorean formulas. Based on these new definitions, we introduce new classes of three-dimensional hyperboloid surfaces and compute their Gauss and mean curvatures. Notably, we demonstrate that these surfaces are geodesic.

The classic Jacobsthal numbers were generalized to k sequences of the generalized order-k Jacobsthal numbers and then have been studied by several authors. In this paper, we explain that all of these studies used an incorrect version of order-k Jacobsthal numbers for reasons and give the correct definition of order-k Jacobsthal numbers. Further, we introduce the compatible generalized order-k Jacobsthal-Lucas numbers with the generalized order-k Jacobsthal numbers. Next, we give some properties of order-k Jacobsthal numbers and order-k Jacobsthal-Lucas numbers, including generating matrix, generalized Binet’s formula, and elementary matrix identities. Further, we investigate specific examples for our results and give special identities, i.e., sum formula and interrelationships between these sequences.