Geometric Interpretations of Keçeci Numbers within Neutrosophic and Hyperreal Number Systems

This study extends the domain of Keçeci Numbers, a unique class of sequences generated by a distinctive algorithm, beyond standard number systems into advanced mathematical structures such as neutrosophic and hyperreal numbers. Keçeci Numbers are sequences produced via a recursive algorithm based on a starting value and an incremental scalar, governed by rules of divisibility and primality. The primary objective of this paper is to investigate the behaviour of this deterministic algorithm within number systems that inherently incorporate the concepts of indeterminacy (neutrosophic) and infinitesimals/infinities (hyperreal), and to interpret these behaviours geometrically. To achieve this, an object-oriented computational framework was first developed using the Python programming language to represent 11 distinct number types, including positive/negative integers, rational, complex, quaternions, neutrosophic, neutrosophic-complex, hyperreal, and bicomplex numbers. Subsequently, the core Keçeci Number algorithm was generalised and redesigned as a unified generator function, ensuring compatibility with all these diverse algebraic structures. This framework enabled the systematic generation of Keçeci sequences within the specified advanced number systems. The main contribution of this work is the geometric analysis of the resulting complex sequences, facilitated by the development of specialised visualisation techniques tailored to each number type. The investigation of Neutrosophic Keçeci Numbers demonstrates the trajectory of the sequence across a plane defined by its determinate and indeterminate components. It is visually demonstrated how the algorithm's "ASK (Augment/Shrink then Check)" rule, applied when prime numbers are encountered, induces sudden and intriguing jumps within this two-dimensional neutrosophic space. In the analysis of Hyperreal Keçeci Numbers, where each number is itself a sequence, the algorithm produces a "sequence of sequences." This high-dimensional behaviour was analysed by plotting the mean values or specific terms of the constituent sequences, revealing underlying trends and patterns that would remain hidden in standard number systems. In conclusion, this research successfully transports the concept of Keçeci Numbers into the neutrosophic and hyperreal domains for the first time, uncovering their geometric and dynamic properties therein. The developed software framework bridges the gap between algorithmic number theory and non-standard analysis, offering a powerful analytical and visual tool for future investigations.

Keywords: Keçeci Numbers, Neutrosophic Numbers, Hyperreal Numbers, Geometric Interpretation, Number Sequences, Algorithmic Generation, Visualisation, Computational Mathematics, Indeterminacy, Non-Standard Analysis.