Gaps Between Integers Having a Common Divisor with an Odd Semiprime

This paper elucidates the distribution law of integers that share a common divisor with an odd semiprime N = pq, where p and q are odd primes satisfying λp < q < (λ+1)p, and λ is a positive integer. It demonstrates that within the interval [1,N −1], the gaps between integers having p or q as a divisor exhibit symmetric behavior ranging from 0 to p −1. Specifically, each gap value from 0 to p −2 appears symmetrically and exactly twice, while the gap value p−1 occurs symmetrically and precisely q− p−1 times across p distinct subintervals. Among these p subintervals, q −λp −1 subintervals each contain λ gaps of value p −1, while the remaining subintervals each contain λ−1 gaps of value p−1.