Unique Properties of the Fibonacci and Lucas Sequences
Preliminary Remarks. Congratulations to the author on his research. The paper is very
interesting and has much potential. It is not quite ready for publication, however. Was it
proofread or revised? The paper doesn’t flow, and there are many typographical,
spelling, and other errors. This reviewer recommends rethinking the structure of the
paper, fixing the errors, and resubmitting what, in the end, may be a beautiful paper.
Specific Comments.
Page 1. In the abstract, the sentence “Using this isomorphism…” is unclear, there should
be semicolons, not commas, in the list in the last sentence, and the word ‘and’ should be
added before (4).
In the Introduction, “that have captivated the world” is hackneyed. Second sentence:
perhaps omit “similar structure compared with the,” and perhaps replace with “have a
structure similar to that of.” Delete the comma after Yang[9]. Next sentence: “take this
isomorphism in consideration”? Do you mean “into” or “under” consideration? Same
sentence: insert “in addition to the work of Horodam [5]” after the word ‘isomorphism’
and delete it from its present location in the sentence. Last sentence: too many uses of
the word ‘specific’—it was distracting.
In the first sentence of the Background section: sequences don’t depend on knowledge.
Next sentence: This is where you should probably define Fibonacci-like sequences, not
the next page.
Page 2. First paragraph. First sentence: isn’t the human body part of nature? Next
sentence: rewrite. Do you mean ‘led to’ or ‘leads to’ (in any case, you don’t men ‘lead
to.’). Last sentence: maybe mention what Lucas pointed out or where he did so? Was he
mentioning the rabbit problem in the Liber Abaci, or something else? Not necessary to
mention, but readers might find it interesting.
Under Definition 2: perhaps use a colon instead of a period for the list? But the whole
sentence needs reworking. Do you mean: “…but the following relations are the ones in
which we are most interested” or “the ones we will find most helpful”?
Section 3: First sentence. If you insist on waiting until now to define Fib-like sequences,
then rewrite this first sentence to something such as: “We will now define and discuss
important properties of Fibonacci-like sequences in terms of recursive sequences and in
terms of 2X2 matrices.” You don’t need “These properties will be discussed,” and you
need to fix the “in 2X2 matrices.” You mean “in terms of,” right? If so, you can even
delete the second ‘in terms of.’
Last sentence before Definition 5: very cool idea, but seems to be in the middle of
nowhere.
In Theorem 1’s proof, which could be written up a tad better, there is a typo: should be
A2 – A1 in both lines (for n = 1 and for n = 2).
Page 3. Example 3: how is everything between Bn = Fn + 8Fn-1 and Definition 6 an
illustration? Of what? Also, may we assume that what makes Definition 6 necessary the
fact you are taking the absolute value? Paragraph between Definition 6 and Theorem 2:
Stick to one number (singular or plural). Change first word from ‘By’ to ‘After,’ and
delete the word ‘can.’ Delete the period between ‘matrix’ and ‘Having,’ insert the word
‘which’ preceded by a comma, and delete the words ‘it is,’ so the sentence now reads:
“After converting a Fibonacci-like sequence into a Fibonacci-like matrix , we take the
determinant of that matrix, which provides us with a way to classify …” and then the rest
of the sentence is unclear. What are you classifying?
Page 4. First sentence: delete “If we consider” and replace “we obtain” with “is.” In the
sentence that begins “Therefore”: delete “since this is true, then” and insert a word of
some sort between “we can” and “in the sequence of numbers.” In Remark 1: Do you
mean to use the word ‘notation’(and not the word ‘notion’)? Theorem 3: The symbols,
upside down A and backwards E, shouldn’t be used, because you are writing in the
language, not about the language. In the proof: there is a typo in the second matrix—it
shouldn’t be Bn -+1. Remark 2: You wrote, “By expressing a sequence in matrix, we
can… .” Do you mean “in matrix form” or “as a matrix”? Same sentence: delete the
‘s’—should be “every element,” not “every elements.” Theorem 4, Proof: Delete “In
order for this to be true.” Also, delete “Afterwards.” Insert ‘that’ between
‘multiplication’ and ‘is associative’ (or else delete the other ‘that’ in the sentence). How
about “there are no zero divisors” instead of the logical symbol?
Page 5. First sentence: How about: “Here we prove there are no zero divisors, leaving
the remainder of the proof to the reader” and deleting the last line? Also, after (-b –rad5
b), there are two commas; delete one of them. Delete the logical symbol,l and replace
with words.
Section 4. Second sentence: change to “..another element in F, which happens because
the Fibonacci… ” (the construction “the reason why...is because…” is considered
ungrammatical.). Also, double-check the details of the proof of Theorem 5.
Page 6. No idea what the first sentence is supposed to read. Please reread it aloud.
Remark 3: The behavior between? Word choice issues. Second sentence is unclear.
Consider rewriting the entire remark. Second to last line on page: “Therefore, we study
when Y =0.” Not a good sentence.
Page 7. Fourth line (“If b = 2a, then…”): Do you mean, “If b = - 2a”? There seems to be
a negative sign missing. Remark 4: change ‘which’ to ‘that.’
Conclusion. First sentence: “can lead in several directions” is better
Acknowledgment. Delete the semicolon, for one thing, but, more important, it seems that
Dr. Jacobson is tacked onto the end. Mention him first. “I would like to thank Dr.
Charlie Jacobson and Elmira College… .”
References. Spelled ‘Quarterly’ wrong at least twice. Shouldn’t Fibonacci, Pell, Lucas,
etc., all be capitalized? Under the third reference, delete ‘edition’ (The word is
erroneously repeated.).
Best wishes for your next version of this fascinating paper!