Some Remarkable Identities Involving Numbers

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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].

References

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

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

  • [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.

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

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

  • [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.

  • [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.

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

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

Formalized Mathematics

(a computer assisted approach)

Journal Information


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

Target Group

researchers in the fields of formal methods and computer-checked mathematics

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