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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

Open Access
Online
ISSN
1898-9934
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Volume 22, Issue 3 (Sep 2014)

Issues

Some Remarkable Identities Involving Numbers

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

Summary

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

References

  • [1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Google Scholar

  • [2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Google Scholar

  • [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

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

  • [6] M.I. Mostafa. A new approach to polynomial identities. The Ramanujan Journal, 8(4): 423-457, 2005. ISSN 1382-4090. doi:10.1007/s11139-005-0272-3.CrossrefGoogle Scholar

  • [7] Werner Georg Nowak. On differences of two k-th powers of integers. The Ramanujan Journal, 2(4):421-440, 1998. ISSN 1382-4090. doi:10.1023/A:1009791425210.CrossrefGoogle Scholar

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

  • [9] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Google Scholar

About the article

Received: 2014-09-05

Published Online: 2014-03-31

Published in Print: 2014-09-01


Citation Information: Formalized Mathematics, 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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