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 multiples of p or q exhibit symmetric behavior ranging from 0 to p - 1. Specifically, each gap from 0 to p - 2 appears exactly twice in a symmetric manner, while the gap p - 1 occurs precisely q - p - 1 times across p distinct subintervals. Among these p subintervals, q - λp - 1 subintervals each contain λ gaps of size p - 1, while the remaining subintervals each contain λ - 1 gaps of size p - 1. From a positional perspective, there are either λ or λ + 1 multiples of p between two adjacent multiples of q, with exactly λ multiples of p appearing both before the first multiple of q and after the last multiple of q. These findings can significantly facilitate the design of algorithms for identifying the divisors of unfactorized composite integers.
Author
(s) Details
Xingbo
Wang
Guangzhou College of Applied Science and Technology, Guangzhou
City, 511370, PRC and Foshan University, Foshan City, 528000, PRC.
Please see the book here:- https://doi.org/10.9734/bpi/mcsru/v4/4832
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