Structural Patterns of Goldbach Partition Numbers: A High-Precision Estimation Model Based on Prime Density

Abstract This research proposes a new approach to the Goldbach Conjecture based on the relationship between the partition numbers of even integers and interval prime density. Through in-depth analysis of even number decomposition properties, we discovered that even numbers can be classified into two categories: Type I (without effective prime factors) and Type II (with effective prime factors). This classification method makes the calculation of even number partition numbers more systematic. In our study, we defined the set X = {x|x = 2n - p, 3 ≤ p ≤ p_m ≤ n}, where p is an odd prime, and through rigorous mathematical proof and large-scale numerical validation, confirmed that there exists a constant ratio (approximately 1.32α) between the prime density in set X and the prime density in the interval [n, 2n]. Based on this discovery, we established an accurate partition number estimation model: G(2n) = c1α nf1f2, where c1 ≈ 1.32 is the Goldbach constant, α is the even number factor coefficient, and f1 and f2 are the prime densities in intervals [3, n] and [n, 2n] respectively. This research conducted comprehensive validation within the range of 1 billion and representative sampling in the 10-100 billion range, showing that the estimation model has high accuracy, with relative errors not exceeding 0.2% for even numbers above 100 million. More importantly, we proved that the G(2n) values for Type I even numbers increase monotonically as n increases, and that partitioning values for Type I even numbers are typically smaller than those for Type II even numbers, thus providing a new path for proving the Goldbach Conjecture. Mathematics Subject Classification (2020): 11P32, 11N05, 11Y35, 11Y16