Diversity of Keçeci Numbers and Their Application to Prešić-Type Fixed-Point Iterations: A Numerical Exploration

Mehmet Keçeci¹

¹ORCID : https://orcid.org/0000-0001-9937-9839, İstanbul, Türkiye

Received: 21.05.2025

Abstract:

This study investigates the role of Keçeci Numbers, as defined by M. Keçeci and encompassing a broad spectrum of number systems (positive/negative integer-like real numbers, floating-point numbers, rational numbers, complex numbers, and quaternions), as a potential number-theoretic exploratory tool and their applicability to Prešić-type fixed-point iterations. Keçeci Numbers were generated via the `kececinumbers` Python module across six defined types using specified algorithmic steps. Each sequence of Keçeci Numbers is also associated with a "Keçeci Prime Number," which may be a characteristic feature of the sequence. The primary objective of this work is to numerically examine how values derived from these diverse Keçeci Number types, when used as the `γ` parameter in a simplified Prešić-type iteration of the form `x<sub>{n+2}</sub> = α*x<sub>n</sub> + β*x<sub>{n+1}</sub> + γ`, influence the convergence behavior of the iteration and the resulting fixed points. For this purpose, Keçeci Numbers generated from different types (processed into real numbers where necessary, e.g., by taking absolute values of complex numbers or real/scalar parts of quaternions) were employed as the source for the `γ` parameter, and iterations were performed. These `γ` parameters were adjusted by a uniform scaling factor to ensure fixed points remained within an observable range. The obtained results demonstrate that the chosen type of Keçeci Number and the `γ` value derived from it directly affect the value of the fixed point to which the Prešić-type iteration converges. For instance, `γ` values derived from positive integer-like Keçeci Numbers or rational numbers led to different fixed-point outcomes compared to those derived from complex numbers (using their absolute values) or quaternions (using their real parts). Notably, a monotonic relationship (in this linear model) was observed between the magnitude of the Keçeci Numbers (or their processed forms) used as the `γ` parameter and the achieved fixed point. Furthermore, iterations were tested using various Keçeci Numbers, including "Keçeci Primes" obtained from specific Keçeci Number sequences, and their fixed-point behaviours were compared. These findings suggest that Keçeci Numbers are not merely abstract mathematical constructs but can also offer a practical toolkit for exploring and understanding the behavior of dynamical systems and iterative processes. The ability to derive Keçeci Numbers from different algebraic structures (rational, complex, quaternion) makes them a flexible resource for parameter selection or system behavior exploration in various mathematical and potential engineering problems. This study lays the groundwork for further investigation into the broader number-theoretic properties of Keçeci Numbers and their potential applications in other mathematical domains.

Keywords:

Keçeci Numbers, Prešić-Type Fixed-Point Theorem, Iterative Methods, Numerical Exploration, Complex Numbers, Quaternions, Rational Numbers, Number Theory, Dynamical Systems, Keçeci Prime Number