Abstract
The paper first puts forward a way to study odd integers by placing the odd integers in a given interval on a perfect full binary tree, then makes an investigation on the odd integers by means of combining the original properties of the integers with the properties of the binary trees and obtains several new results on how an odd integer's divisors distribute on a level of a binary tree. The newly discovered law of divisors' distribution that includes common divisors between two symmetric nodes, genetic divisors between an ancestor node and its descendant node can provide a new and simple approach to factorize odd composite integers. Based on the mathematical deductions, numerical experiments are designed and demonstrated in the Maple software. All the results of the experiments are conformance to expectation and validate the validity of the approach.
Highlights
In 2016, WANG X in article (WANG X, 2016(IJSIMR)) put forward an approach that studies integer by putting odd integers bigger than 1 on a full perfect binary tree from the top to the bottom and from the left to right
This approach derived out many previously-unknown properties of the odd integers, such as properties of symmetric nodes and symmetric common divisors, properties of subtrees’ duplication and transition, and properties of sum by level, root division and the genetic traits, as introduced in WANG’s articles (WANG X, 2017(JM), 2017(GJPAM),2019(IJAPM))
These new properties could be helpful in solving the problem of integer factorization, as probed in FU’s paper (FU D,2017(JCE)),WANG’s paper(WABG X, 2017(JCE)) and LI’s paper (LI J., 2018(AJCM)), and they would be helpful for knowing of the RSA modulus, as investigated in papers (WANG X, 2018(JMR), WANG X. 2018(IJMSS))
Summary
In 2016, WANG X in article (WANG X, 2016(IJSIMR)) put forward an approach that studies integer by putting odd integers bigger than 1 on a full perfect binary tree from the top to the bottom and from the left to right. This approach derived out many previously-unknown properties of the odd integers, such as properties of symmetric nodes and symmetric common divisors, properties of subtrees’ duplication and transition, and properties of sum by level, root division and the genetic traits, as introduced in WANG’s articles (WANG X, 2017(JM), 2017(GJPAM),2019(IJAPM)). This paper introduces the new approach and its traits in factoring odd composite integers
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