A Unified Approximation Formula for Prime Numbers denisty

This research presents the discovery of the Adith Sequence, a recursive mathematical progression designed to model the distribution of prime numbers with high precision. By leveraging the unique growth properties of this sequence, I derive the Unified Approximation Formula (UAF), a new analytical model for estimating the prime-counting function π(x). A critical feature of this work is the identification of a Universal Constant inherent to the sequence’s convergence. This constant serves as the foundation for the UAF, allowing for a significant reduction in the error margin compared to the classical Prime Number Theorem (PNT). Comparative analysis across magnitudes up to 10⁵⁰ demonstrates that the UAF maintains a tighter fit to the true prime count than traditional models. This paper details the recursive definition of the sequence, the derivation of the UAF, and a comprehensive error margin analysis, offering a fresh perspective on the underlying density of prime numbers