Empirical modeling of the distribution of prime numbers and fixed initial digit twins

This study proposes an empirical and computational approach to modeling the distribution of prime numbers and identifying a subset of twin primes with constrained initial digits. Starting from the function \( p_2 = 2p_1 - z \), we investigate the behavior of the variable \( z \) and its relation to the gap between consecutive primes. Our numerical analysis suggests that the ratio \( x = \frac{z}{p_1} \) tends to 1 as \( p_1 \) increases, revealing a hidden asymptotic structure. We also present a parametric function inspired by Willans’ formula to predict the positions of twin primes beginning with a fixed digit, and introduce a generalized version applicable to other digit constraints. Experimental tests show that the proposed formulas consistently detect twin primes of increasing length starting with digits like 2 and 3. Although our approach is not derived from a formal proof, it highlights potential regularities in the apparent chaos of prime distribution and suggests that an infinite number of structurally detectable twin primes may exist.