New Discoveries on Integer Factorization by Valuated Binary Tree

Abstract

The paper proves that an odd composite integer N can be factorized in O(logN) time if N's big divisor is in the form 2α u + 1 or 2α u − 1 and its small divisor does not exceed 2α + 1 Theorems and corollaries around the conclusion are proved with detail mathematical reasoning. The results in the paper again demonstrate the significance of applying the valuated binary tree on analyzing the odd integers.