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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

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Volume 24, Issue 3 (Sep 2016)

Issues

Prime Factorization of Sums and Differences of Two Like Powers

Rafał Ziobro
  • Department of Carbohydrate Technology University of Agriculture Krakow, Poland
Published Online: 2017-02-21 | DOI: https://doi.org/10.1515/forma-2016-0015

Abstract

Representation of a non zero integer as a signed product of primes is unique similarly to its representations in various types of positional notations [4], [3]. The study focuses on counting the prime factors of integers in the form of sums or differences of two equal powers (thus being represented by 1 and a series of zeroes in respective digital bases).

Although the introduced theorems are not particularly important, they provide a couple of shortcuts useful for integer factorization, which could serve in further development of Mizar projects [2]. This could be regarded as one of the important benefits of proof formalization [9].

MSC: 11A51; 03B35

Keywords: integers; factorization; primes

MML: identifier: NEWTON03; version: 8.1.05 5.37.1275

References

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  • [2] Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi: [Crossref] [Crossref]

  • [3] Paul Erdős and János Surányi. Topics in the Theory of Numbers, chapter Divisibility, the Fundamental Theorem of Number Theory, pages 1-37. Springer New York, 2003. doi: [Crossref] [Crossref]

  • [4] Jacek Gancarzewicz. Arytmetyka, 2000. In Polish.

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  • [6] Artur Korniłowicz and Piotr Rudnicki. Fundamental Theorem of Arithmetic. Formalized Mathematics, 12(2):179-186, 2004.

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  • [8] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990.

  • [9] Adam Naumowicz. An example of formalizing recent mathematical results in Mizar. Journal of Applied Logic, 4(4):396-413, 2006. doi: [Crossref] [Crossref] Towards Computer Aided Mathematics.

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  • [13] Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.

  • [14] Rafał Ziobro. Fermat’s Little Theorem via divisibility of Newton’s binomial. Formalized Mathematics, 23(3):215-229, 2015. doi: [Crossref] [Crossref]

About the article

Received: 2016-06-30

Published Online: 2017-02-21

Published in Print: 2016-09-01



Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, DOI: https://doi.org/10.1515/forma-2016-0015. Export Citation

© by Rafał Ziobro. This work is licensed under version 3.0 of the Creative Commons Attribution–ShareAlike License (CC BY-SA 3.0 LEGALCODE)

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