Abstract
The article makes an investigation on the probability of finding the greatest common divisor between a given integer and a hidden integer that lies in an integer interval. It shows that, adding the integers that are picked randomly in the interval results in a much bigger probability than subtracting the picked integers one with another. Propositions and theorems are proved and formulas to calculate the probabilities are presented in detail. The research is helpful in developing probabilistic algorithm of integer factorization.
Highlights
The article makes an investigation on the probability of finding the greatest common divisor between a given integer and a hidden integer that lies in an integer interval
Given an odd integer N that has the greatest common divisor (GCD) d with another odd integer e in a large odd interval that consists in consecutive odd numbers, what is the probability to find out d ? This question is close to the problem of integer factorization, as investigated in (Xingbo Wang, 2017(1)), (Jianhui Li, 2017),(Dongbo Fu, 2017) and (Xingbo Wang, 2017(2))
Proposition 2 Suppose N is a composite odd integer and n is a positive integer; let S = {s1, s{2, · · ·, sn} be a set that consists in n consecutive odd integers that satisfy, for an m with 1 ≤ m ≤ n, GCD(N, si) =
Summary
Given an odd integer N that has the greatest common divisor (GCD) d with another odd integer e in a large odd interval that consists in consecutive odd numbers, what is the probability to find out d ? This question is close to the problem of integer factorization, as investigated in (Xingbo Wang, 2017(1)), (Jianhui Li, 2017),(Dongbo Fu, 2017) and (Xingbo Wang, 2017(2)). As stated in (Xingbo Wang, 2017(3)) the answer to the question relies on a detail study on the properties of odd integers on an odd interval. This article makes an investigation on the probability of finding the integer e in a large odd interval. The research shows that, by adding two or more terms contained in the interval, it has a very big probability to find out d
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