New Constructive Approach To Solve Problems of Integers' Divisibility
Keywords:
Mathematical proof, Elementary number theory, Divisibility, Consecutive Integers, Residue system, Constructive Proof.Abstract
This paper aims at introducing a new constructive approach to solve problems in elementary number theory. It starts with a comprehensive analysis on present approaches to solve problems related with divisible features of consecutive integers, which include consecutive positive integers, consecutive positive odd integers and consecutive positive even integers; then it detailly demonstrates advantages and disadvantages of the present-applied approaches in their deducing process, especially the conflicts in proving the almost same-stated statements; in the end the paper puts forward a new constructive approach and uses it to have a new proof for the three fundamental theorems: for any positive integer n and among n consecutive positive integers there exists one and only one that can be divisible by n; for any positive odd integer p and among p consecutive positive odd integers there exists one and only one that can be divisible by p; for a positive even integer w and among w consecutive positive even integers, there exist exactly two that can be divisible by w. The new constructive proof is valuable for more extensive utilities in elementary number theory.
References
Kenneth H Rosen. Elementary Number Theory and Its Application.Pearson Education, Inc., 2005.
Melvyn B Nathanson. Elementary Methods in Number Theory.Springer, 2000
Daniel Sutantyo. Elementary and Analytic Methods in Number Theory. Thesis for the degree of Master of Science, Macquarie University,August 2007
Gary Chartrand, Albert D. Polimeni, Ping Zhang. Mathematical Proofs: A Transition to Advanced Mathematics.Pearson Education, Limited, 2012-9-17
Peter J. Eccles. An Introduction to Mathematical Reasoning. London:Cambridge University Press, 1997-12-11
Antonena Cupilari. The Nuts and Bolts of Proofs,ELSEVIER ACADEMIC PRESS(Third Edition),2005
Martin Sleziak. Prove that one of n consecutive integers must be divisible by n. TechQues,http://www.techques.com/question/29-126766/Prove-that-one-of-n-consecutive-integers-must-be-divisible-by-n,2012-04-01
Martin Sleziak. Prove that one of n consecutive integers must be divisible by n. Stack Exchange,http://math.stackexchange.com/questions/126766/prove-that-one-of-n-consecutive-integers-must-be-divisible-by-n,2012-04-01
Cryptonym Author. Want a Proof for the statement that one and only one of n consecutive integers must be divisible by n. Baiduzhidao(in Chinese),http://zhidao.baidu.com/link? url=U-CjBha1E-KVQdCZXNuJHHkEegDfzupkZxN4ow8aCjFa3ox4nFAu0QqU5KFYV1AXv-XOPkxvnz8jJyHoM-qu8a, 2012-08-11
liuxiaof. Please prove that one and only one of n consecutive integers must be divisible by n. Baiduzhidao(in Chinese),http://zhidao.baidu.com/link?url=XCYzpMK-9aVIvv1QrrDcRor9-ejy_JB1xApSbbze1PXXwzrH6mFGOCJp7mwxtbHVt9YbMNC1QvxMdQunderlineDqtwqxa,2013-09-05
orangedom. Need to prove that one and only one of n consecutive integers must be divisible by n. Baiduzhidao(in Chinese), http://zhidao.baidu.com/link?url=sjnJnm7y_lYXhtXuLYnympIq TM0g77r69O7kR4dUr3rRCnCcGR1qyvnD4pSbvo1eBwcn2uOlL8GQvwrM2a5W_,2006-06-10
ImAGinE. Use Modulo to prove that one and only one of n consecutive integers must be divisible by n. Baiduzhidao(in Chinese), http://zhidao.baidu.com/link?url=3hZrAUE0u6P4QyhzPegvJUkQYABbwenruydadyUYgsVxp_4K87_5MSUPIBW3aNtbUEkXNqypN_MBq68VasQeU_,2009-06-01
AMIN WITNO.Number thoery,Philadelphia University,http://www.philadelphia.edu.jo/math/witno/notes/number_theory.pdf,2007
Crystalyxh. Prove that one and only one of n(n>1) consecutive integers must be divisible by n. Baiduzhidao(in Chinese), http://zhidao.baidu.com/link?url=O0KChBx-GNpbkSOukeU-fsSwoKVTfJVdTCAuv8VQ0VhNHfqJDObXi4zSCNehiyawNp2Lqql_gpmwr-C91wGRba, 2009-10-12
Shanshuiluqiao. Answer to a question. SinaAsk(in Chinese), http://iask.sina.com.cn/b/ 20497807.html, 2012-07-26
Downloads
Published
Issue
Section
License
- Papers must be submitted on the understanding that they have not been published elsewhere (except in the form of an abstract or as part of a published lecture, review, or thesis) and are not currently under consideration by another journal published by any other publisher.
- It is also the authors responsibility to ensure that the articles emanating from a particular source are submitted with the necessary approval.
- The authors warrant that the paper is original and that he/she is the author of the paper, except for material that is clearly identified as to its original source, with permission notices from the copyright owners where required.
- The authors ensure that all the references carefully and they are accurate in the text as well as in the list of references (and vice versa).
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Attribution-NonCommercial 4.0 International that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
- The journal/publisher is not responsible for subsequent uses of the work. It is the author's responsibility to bring an infringement action if so desired by the author.
