A New Algorithmic Approach to Integer Divisibility and Factorization

Abstract

This article proposes an algorithmic method for testing divisibility, grounded in the relationships between the multiplication tables of consecutive divisors. The algorithm generates, through an iterative process, a sequence of quotients and differences that makes detecting divisibility easier. This approach also offers a representation of odd integers as alternating sums of multiples, paving the way for the formulation of the Kadouno—Serre conjecture, which concerns the convergence of alternating series. The algorithm is then extended to pairs of non-consecutive divisors, along with a mathematical study of its convergence. Finally, a hybrid version—combining this method with an optimized initialization of Fermat’s factorization—is proposed to improve efficiency on balanced semiprimes. Potential applications are outlined, notably in pedagogy and number theory.