Tabular Arithmetic and Deterministic Primality Testing via Multiplication Tables.

Introduction. Multiplication tables, traditionally used in foundational mathematics education, also harbor rich arithmetic structures when analyzed through operations of sum, difference, and product. This study introduces a novel, deterministic prime-detection methodology derived from the structural patterns in these tables, adapted for large numbers via a logarithmic framework.

Methods. We formalize three operations on the multiplication table T(n) of a fixed integer n: the tabular sum (∑), difference (δ), and product (∏). We derive closed-form expressions for ∑ (linking to triangular numbers), δ (relating to arithmetic sequences), and ∏ (corresponding to factorial constructs), leading to the Kadouno Primality Test. A logarithmic adaptation exploits Stirling’s approximation and a dynamic scaling regulated by decimal thresholds to maintain computational tractability for very large integers. Further refinement replaces the factorial structure with a primorial-based test that preserves small-prime exclusion and enhances algorithmic efficiency.

Results. The primality test reliably distinguishes prime from composite numbers across multiple ranges. With the logarithmic version, the method remains accurate and efficient even as input size grows, thanks to a dynamically adjusted depth parameter. Incorporating the primorial variant secures robust rejection of Carmichael numbers—a significant advantage over probabilistic Fermat-type tests.

Conclusions. This research presents a deterministic, scalable, and pedagogically intuitive primality test grounded in elementary tabular analysis. Its dual utility in both computational number theory and mathematics education positions it as a valuable contribution with practical and theoretical impact. Future work will explore integration with educational software and further extension to cryptographic applications.