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Formulating an Odd Perfect Number: An in Depth Case Study

Received: 17 September 2018     Accepted: 29 October 2018     Published: 30 November 2018
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Abstract

A perfect number is a positive integer that is equals to the sum of its proper divisors. No one has ever found an odd perfect number in the field of Number Theory. This paper review discussed the history and the origin of Odd Perfect Numbers. The theorems and proofs are given and stated. This paper states the necessary conditions for the existence of odd perfect numbers. In addition, several related studies such as “Odd Near-Perfect Numbers” and “Deficient-Perfect Numbers”. Formulating odd perfect numbers will have a significant contribution to other Mathematics conjectures. This paper compiles all the known information about the existence of an odd perfect number It also lists and explains the necessary theorems and lemmas needed for the study. The results and conclusions shows the ff: Odd Perfect Numbers has a lower bound of 101500, The total number of prime factors/divisors of an odd perfect number is at least 101, and 108 is an appropriate lower bound for the largest prime factor of an odd perfect number and the second large stand third largest prime divisors must exceed 10000 and100 respectively. In summary, it found out that there is a chance for an odd perfect number to exist even if there is a very small possibility.

Published in Pure and Applied Mathematics Journal (Volume 7, Issue 5)
DOI 10.11648/j.pamj.20180705.11
Page(s) 63-67
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2018. Published by Science Publishing Group

Keywords

Perfect Number, Odd, Theorem, Number Theory

References
[1] Perfect number. (2018). Retrieved from https://en.wikipedia.org/wiki/Perfect_number.
[2] Perfect Numbers. Retrieved from https://math.tutorvista.com/number-system/perfect-numbers.html.
[3] Bogomolny, Alexander. (2017) Pythagorean Triples and Perfect Numbers.
[4] Mersenne Prime. (2018). Retrieved from https://en.wikipedia.org/wiki/Mersenne_prime.
[5] Caldwell, Chris K. (2016). Mersenne Primes: History, Theorems and Lists. Retrieved from https://primes.utm.edu/mersenne/.
[6] Brown, D. R. (2011). "Odd Perfect Numbers".
[7] Carr, Aver. (2013). "Odd Perfect Numbers: Do They Exist?"
[8] The Largest Perfect Number (2016, February 7). Retrieved from http://www.mathnasium.com/www-mathnasium-com-fremont-news-the-largest-erfect-number.
[9] Review Papers. (2001). Retrieved from http://websites.uwlax.edu/biology/ReviewPapers.html.
[10] Bhat, Adi. (2018). http://libguides.usc.edu/writingguide/qualitative.
[11] What is the difference between a research paper and review paper. (2015). Retrieved from https://www.editage.com/insights/what-is-the-difference-between-a-research-paper-and-a-review-paper.
[12] Sarmiento, Janilo. (2012). Retrieved from https://www.slideshare.net/janilosarmiento/the-ipo-model-of-evaluation-input, process, output.
[13] Stevens, Ben. (2012). A Study on Necessary Conditions for Odd Perfect Numbers”.
[14] Adajar, Carlo. (2016, University of the Philippines Diliman), On Odd Near-Perfect and Deficient- Perfect Numbers.
[15] Almost Perfect Number. (2018). Retrieved from https://en.wikipedia.org/wiki/Almost_perfect_number.
Cite This Article
  • APA Style

    Renz Chester Rosales Gumaru, Leonida Solivas Casuco, Hernando Lintag Bernal Jr. (2018). Formulating an Odd Perfect Number: An in Depth Case Study. Pure and Applied Mathematics Journal, 7(5), 63-67. https://doi.org/10.11648/j.pamj.20180705.11

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    ACS Style

    Renz Chester Rosales Gumaru; Leonida Solivas Casuco; Hernando Lintag Bernal Jr. Formulating an Odd Perfect Number: An in Depth Case Study. Pure Appl. Math. J. 2018, 7(5), 63-67. doi: 10.11648/j.pamj.20180705.11

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    AMA Style

    Renz Chester Rosales Gumaru, Leonida Solivas Casuco, Hernando Lintag Bernal Jr. Formulating an Odd Perfect Number: An in Depth Case Study. Pure Appl Math J. 2018;7(5):63-67. doi: 10.11648/j.pamj.20180705.11

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  • @article{10.11648/j.pamj.20180705.11,
      author = {Renz Chester Rosales Gumaru and Leonida Solivas Casuco and Hernando Lintag Bernal Jr},
      title = {Formulating an Odd Perfect Number: An in Depth Case Study},
      journal = {Pure and Applied Mathematics Journal},
      volume = {7},
      number = {5},
      pages = {63-67},
      doi = {10.11648/j.pamj.20180705.11},
      url = {https://doi.org/10.11648/j.pamj.20180705.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20180705.11},
      abstract = {A perfect number is a positive integer that is equals to the sum of its proper divisors. No one has ever found an odd perfect number in the field of Number Theory. This paper review discussed the history and the origin of Odd Perfect Numbers. The theorems and proofs are given and stated. This paper states the necessary conditions for the existence of odd perfect numbers. In addition, several related studies such as “Odd Near-Perfect Numbers” and “Deficient-Perfect Numbers”. Formulating odd perfect numbers will have a significant contribution to other Mathematics conjectures. This paper compiles all the known information about the existence of an odd perfect number It also lists and explains the necessary theorems and lemmas needed for the study. The results and conclusions shows the ff: Odd Perfect Numbers has a lower bound of 101500, The total number of prime factors/divisors of an odd perfect number is at least 101, and 108 is an appropriate lower bound for the largest prime factor of an odd perfect number and the second large stand third largest prime divisors must exceed 10000 and100 respectively. In summary, it found out that there is a chance for an odd perfect number to exist even if there is a very small possibility.},
     year = {2018}
    }
    

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    T1  - Formulating an Odd Perfect Number: An in Depth Case Study
    AU  - Renz Chester Rosales Gumaru
    AU  - Leonida Solivas Casuco
    AU  - Hernando Lintag Bernal Jr
    Y1  - 2018/11/30
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    N1  - https://doi.org/10.11648/j.pamj.20180705.11
    DO  - 10.11648/j.pamj.20180705.11
    T2  - Pure and Applied Mathematics Journal
    JF  - Pure and Applied Mathematics Journal
    JO  - Pure and Applied Mathematics Journal
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    EP  - 67
    PB  - Science Publishing Group
    SN  - 2326-9812
    UR  - https://doi.org/10.11648/j.pamj.20180705.11
    AB  - A perfect number is a positive integer that is equals to the sum of its proper divisors. No one has ever found an odd perfect number in the field of Number Theory. This paper review discussed the history and the origin of Odd Perfect Numbers. The theorems and proofs are given and stated. This paper states the necessary conditions for the existence of odd perfect numbers. In addition, several related studies such as “Odd Near-Perfect Numbers” and “Deficient-Perfect Numbers”. Formulating odd perfect numbers will have a significant contribution to other Mathematics conjectures. This paper compiles all the known information about the existence of an odd perfect number It also lists and explains the necessary theorems and lemmas needed for the study. The results and conclusions shows the ff: Odd Perfect Numbers has a lower bound of 101500, The total number of prime factors/divisors of an odd perfect number is at least 101, and 108 is an appropriate lower bound for the largest prime factor of an odd perfect number and the second large stand third largest prime divisors must exceed 10000 and100 respectively. In summary, it found out that there is a chance for an odd perfect number to exist even if there is a very small possibility.
    VL  - 7
    IS  - 5
    ER  - 

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Author Information
  • Senior High School Department, Arellano University, Pasay, Philippines

  • Senior High School Department, Norzagaray National High School, Norzagaray, Philippines

  • General Education Department, Far Eastern University – NRMF, Fairview Quezon City, Philippines

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