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The Prime Density Triangle offers a novel combinatorial framework for exploring properties of prime numbers. From the algebraic relations embedded within this triangle, 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 deliberate construction—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 Den- sity Triangle’s algebraic frameworks, rigorous convergence proofs including asymptotic error 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.

Square root approximation, Prime Density Triangle, Exponential-polynomial means, Non-iterative methods, Asymptotic analysis, Algebraic relations.