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

This article introduces an innovative arithmetic framework based on the analysis of multiplication tables through three fundamental operations: the tabular sum, the tabular difference, and the tabular product. These operations reveal deep numerical structures that lead to the construction of an original primality test. The method relies on a combinatorial formulation of products derived from multiplication tables, with a logarithmic optimization designed for large integers. An adaptive function dynamically adjusts the test depth according to the size of the number under analysis. A variant based on the primorial is also introduced, allowing refined detection of composite numbers, including Carmichael numbers. Numerous examples demonstrate the effectiveness of this approach on a wide range of inputs, from small primes to very large integers. This method opens promising perspectives in cryptography, algorithmic number theory, and mathematics education.