Abstract—The paper investigates some divisibility traits on valuated binary trees, including some divisibility rules on T₃ tree and some laws of multiples’ distribution on a T_p tree. It proves that the root of a valuated binary tree can always have a common divisor with certain nodes that appear periodically on the left most path or the left side-path of the tree and calculation of divisors of a node is highly related with the two’s power plus one and the two’s power minus one. Theorems and corollaries are proved with detail mathematical deductions and they provide a foundation for people to design algorithm for factoring odd integers.

Index Terms—Number theory, integer factorization, congruence equation, binary tree.

Xingbo Wang is with Department of Mechatronic Engineering, Foshan University, Foshan City, 528000, China. He is also with Guangdong Engineering Center of Information Security for Intelligent Manufacturing System, Foshan, China and State Key Laboratory of Mathematical Engineering and Advanced Computing, Wuxi, China (email: xbwang@fosu.edu.cn;dr.xbwang@qq.com).
Hongqiang Guo is with Department of Mechatronic Engineering, Foshan University, Foshan City, 528000, China.

Cite: Xingbo Wang, Hongqiang Guo, "Some Divisibility Traits on Valuated Binary Trees," International Journal of Applied Physics and Mathematics vol. 9, no. 1, pp. 1-11, 2019.