Analysis of Multiplication Tables by Sum, Difference and Product: A New Approach to Primality

This paper introduces a novel deterministic framework for the analysis of multiplication tables through three fundamental operations: tabular sum, tabular difference, and tabular product. While multiplication tables are traditionally regarded as elementary pedagogical tools, we demonstrate that they encode deep arithmetic structures when studied systematically. In particular, the tabular product leads naturally to a new approach to primality testing based on factorial and primorial constructions.

We formalize the tabular product as a structured product of table rows and show that its interaction with the greatest common divisor yields a reliable criterion for distinguishing prime and composite integers. To address computational limitations inherent to large factorials, a logarithmic optimization is introduced, relying on asymptotic approximations and adaptive depth control. This refinement allows the method to scale efficiently to very large integers without loss of determinism.

Furthermore, a primorial-based variant of the test is developed, eliminating redundant composite factors and significantly strengthening the detection of Carmichael numbers, which commonly evade classical probabilistic tests. A decimal regulation function is proposed to dynamically adjust the test depth as a function of the size of the integer under consideration.

Theoretical results are supported by explicit numerical examples, illustrating robustness, scalability, and algorithmic relevance. This work opens new perspectives for primality testing, cryptographic applications, and the foundational reinterpretation of elementary arithmetic structures.