<abstract xmlns="http://www.w3.org/1999/xhtml">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].</abstract>

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].

Some Remarkable Identities Involving Numbers

Department of Carbohydrate Technology University of Agriculture Krakow, Poland

Formalized Mathematics

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

{"article-title":"Some Remarkable Identities Involving Numbers"}

The article focuses on simple identities found for binomials, their divisibility, and basic inequalities. A general formula allowing factorization of the sum...

Some Remarkable Identities Involving Numbers

Published Online: Mar 31, 2014

Page range: 205 - 208

Received: Sep 05, 2014

DOI: https://doi.org/10.2478/forma-2014-0023

© by Rafał Ziobro

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