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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year

SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

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Volume 22, Issue 3


Some Remarkable Identities Involving Numbers

Rafał Ziobro
Published Online: 2014-03-31 | DOI: https://doi.org/10.2478/forma-2014-0023


The article focuses on simple identities found for binomials, their divisibility, and basic inequalities. A general formula allowing factorization of the sum of like powers is introduced and used to prove elementary theorems for natural numbers.

Formulas for short multiplication are sometimes referred in English or French as remarkable identities. The same formulas could be found in works concerning polynomial factorization, where there exists no single term for various identities. Their usability is not questionable, and they have been successfully utilized since for ages. For example, in his books published in 1731 (p. 385), Edward Hatton [3] wrote: “Note, that the differences of any two like powers of two quantities, will always be divided by the difference of the quantities without any remainer...”.

Despite of its conceptual simplicity, the problem of factorization of sums/differences of two like powers could still be analyzed [7], giving new and possibly interesting results [6].

Keywords : identity; divisibility; inequations; powers


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  • [3] E. Hatton. An intire system of Arithmetic: or, Arithmetic in all its parts. Number 6. Printed for G. Strahan, 1731. http://books.google.pl/books?id=urZJAAAAMAAJ.Google Scholar

  • [4] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.Google Scholar

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About the article

Received: 2014-09-05

Published Online: 2014-03-31

Published in Print: 2014-09-01

Citation Information: Formalized Mathematics, Volume 22, Issue 3, Pages 205–208, ISSN (Online) 1898-9934, DOI: https://doi.org/10.2478/forma-2014-0023.

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

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