A New Factorial Test for Primality Based on Tabular Arithmetic

This article introduces a novel and elementary primality test based on factorial divisibility and tabular arithmetic. The core idea is to define a function on integers n > 1, using k = square root of n, and check whether gcd(k!, n) = 1. This simple condition yields a deterministic test: it accepts all prime numbers and rejects all composite numbers, including Carmichael numbers, which often defeat classical probabilistic tests such as Fermat’s. The article provides a formal proof of the test’s validity, a justification via the structure of multiplication tables, and a comparative analysis of its rejection power on known Carmichael numbers. It also includes a logarithmic optimization using Stirling’s approximation, an extension via primorial functions, and experimental validation up to the billion scale. This test offers a new characterization of primality rooted in tabular arithmetic and provides an intuitive and efficient alternative to classical methods, with potential applications in education, number theory, and algorithmic research.