JP Journal of Algebra, Number Theory and Applications
Volume 41, Issue 1, Pages 95 - 119
(January 2019) http://dx.doi.org/10.17654/NT041010095 |
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THE USE OF CRAGMONT FIBONACCI MATRICES IN ANALYZING AND CATALOGING IDENTITIES WITH POWERS OF FIBONACCI AND LUCAS NUMBERS
George A. Hisert
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Abstract: This paper discusses identities which involve powers p of Fibonacci and Lucas numbers where p > 2. Among other things, it introduces the “Cragmont Fibonacci Matrix” which can be used to derive and prove any uniform power identity. The Cragmont Fibonacci Matrix also provides a unique set of numbers for each uniform power identity, thus allowing such identities to be easily cataloged. The methodology used in the paper may preempt much of the current literature on the topic of the derivation and proof of identities of order greater than 2. The paper also conjectures that there are no identities with terms of power p except uniform power identities (or identities that can be transformed into uniform power identities). This broad coverage of identities of power p for Fibonacci sequences holds promise for the analysis of order r recurrence sequences where r > 2. |
Keywords and phrases: Fibonacci, Lucas, identity, uniform power, recurrence sequence, Melham’s conjecture, uniform power function, uniform power sum, uniform power identity, UPI, UPF, Lucasian twin, Lucas sequence, non-uniform power identity, Morgado’s identity, Cassini’s formula, Cragmont Fibonacci matrix, Cragmont matrix.
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