SCALE-INVARIANT PROPERTIES IN PRIME NUMBER DISTRIBUTION: A COMPREHENSIVE EMPIRICAL ANALYSIS WITH DETAILED NUMERICAL RESULTS

This paper presents a comprehensive empirical investigation into prime number distribution patterns using the Sieve of Eratosthenes algorithm. We analyze 805 consecutive intervals of 10 numbers each, ranging from 2 to 9,981, computing the ratio of prime sums to non-prime sums for each interval. Our analysis reveals three major findings: Scale-Invariant Property: Intervals with the same prime count maintain nearly constant ratios, with a coefficient of variation below 1%, regardless of magnitude, spanning numbers from the hundreds to the thousands. Quasi-Linear Scaling: The relationship between prime count and ratio demonstrates a power-law behavior, with multiplicative factors of 1.00×, 2.25×, 3.86×, and 6.81× for intervals containing 1, 2, 3, and 5 primes, respectively. Logarithmic Decline: A 13.36% decrease in the mean ratio across the dataset confirms the Prime Number Theorem's prediction of declining prime density. Complete numerical results for all 805 intervals are provided, demonstrating exceptional consistency within prime-count groups. The coefficient of variation remains below 1% for all groups—0.32% for 1 prime, 0.72% for 2 primes, and 0.90% for 3 primes—suggesting fundamental mathematical constraints governing local prime distribution.