EXPONENTIAL-POLYNOMIAL ROOT MEANS DERIVED FROM THE PRIME DENSITY TRIANGLE: AN ELEMENTARY APPROACH TO √m
Year: 2026
Type: preprint
Abstract: The Prime Density Triangle offers a novel combinatorial framework for exploring properties of prime numbers. From the algebraic relations embedded within thistriangle, specifically the coupled expressions X^p1± p1and X^p0± p0, we derive two closed form sequences.First, the multiplication of these expressions yields the sequence Sn(m) = (mn+ n2)1/(n+2), which converges to m. Second, by a deliberateconstruction—replacing the additive denominator n + 2 with the multiplicative denominator 2n—we obtain the Exponential-Polynomial Root Mean (EPRM) defined by Rn(m) = (mn+n2)1/(2n)for m > 0 and n ∈ N. This construction provides a closed form, non-iterative method for approximating √m for sufficiently large n. In this paper, we present a complete derivation of both sequences from the Prime Density Triangle’s algebraic frameworks, rigorous convergence proofs including asymptoticerror analysis, detailed worked examples for m = 2, 3, 5 with explicit computations,numerical summary tables, explicit iterations for classical methods (Babylonian and Newton-Raphson), and a comprehensive comparison. The approach is elementary and highlights the dominance of exponential terms over polynomial corrections.
Author: Batu J J, Yagah
Institution: Ghana Education Service
Language: enm
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204151968