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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 19. Выпуск 2
УДК 510 DOI 10.22405/2226-8383-2018-19-2-259-271
Характеризация чисел Фибоначчи
Пирилло Джузеппе — Кафедра математики и информатики 'У. Диии, Университет Флоренции, Флоренция, Италия. e-mail: [email protected]
Аннотация
В согласии с философско-математической мыслью ранних пифагорейцев, для заданных отрезков s и i мог быть найден отрезок и, содержащийся ровно п раз в s и т раз в t при некоторых подходящих числах п и т. Справедливость этого положения была подвергнута самими же пифагорейцами при обнаружении ими несоизмеримости стороны и диагонали правильного пятиугольника. Это фундаментальное историческое открытие, прославившие Пифагорейскую школу, оставило «забытым» предшествующий ему этап исследований. Именно фаза поиска и, начатая многочисленными неудачными попытками и завершившаяся разработкой известной техники доказательства «чётное-нечётное», является объектом нашей «творческой интерпритации» исследований Пифагора, которую мы приводим в этой статье. В частности, будет выявлена сильная связь между пифагорейским тождеством 6(6 + а) — а2 = 0 относительно стороны 6 и диагонали а правильного пятиугольника и тождеством Кассннн F*Fi+2 — F?+1 = ( — 1)* для трех последовательных чисел Фибоначчи. Более того, эти два тождества были обнаружены Пифагорейской школой «почти одновременно», и, следовательно, числа Фибоначчи и тождество Кассини имеют пифагорейское происхождение. Нам не известны архивные документы (уже столь редкие для изучаемого периода!), касающиеся этого утверждения, но в статье приводятся ряд математических заключений в его подтверждение. Приведенный в работе анализ дает новую (и естественную) характеризацию чисел Фибоначчи, до сих пор отсутствующую в литературе.
Ключевые слова: несоизмеримость, золотое сечение, числа Фибоначчи.
Библиография: 24 названий.
Для цитирования:
Д. Пирилло. Характеризация чисел Фибоначчи // Чебышевский сборник. 2018. Т. 19, вып. 2, С. 259-271.
CHEBYSHEVSKII SBORNIK Vol. 19. No. 2
UDC 510 DOI 10.22405/2226-8383-2018-19-2-259-271
A characterization of Fibonacci numbers
Pirillo Giuseppe Dipartimento di Matematica ed Informatica "U. Dini", Universita di Firenze,
Florence, Italy.
e-mail: [email protected]
Abstract
For the early Pythagoreans, in perfect agreement with their philosophical-mathematical thought, given segments s and t there was a segment u contained exactly n times in s and m times in t, for some suitable integers n and m. In the sequel, the Pythagorean system is been put in crisis by their own discovery of the incommensurability of the side and diagonal of a regular pentagon. This fundamental historical discovery, glory of the Pythagorean School, did however "forget' the research phase that preceded their achievement. This phase, started with numerous attempts, all failed, to find the desired common measure and culminated with the very famous odd even argument, is precisely the object of our " creative interpretation" of the Pythagorean research that we present in this paper: the link between the Pythagorean identity b(b + a) — a2 = 0 concerning the side b and the diagonal a of a regular pentagon and the Cassini identity FiFi+2 — F2+1 = ( — 1)% concerning three consecutive Fibonacci numbers, is very strong. Moreover, the two just mentioned equations were " almost simultaneously discovered by the Pythagorean School and consequently Fibonacci numbers and Cassini identity are of Pythagorean origin. There are no historical documents (so rare for that period!) concerning our audacious thesis, but we present solid mathematical arguments that support it. These arguments provide in any case a new (and natural!) characterization of the Fibonacci numbers, until now absent in literature
Keywords: incommensurability, golden ratio, Fibonacci numbers.
Bibliography: 24 titles.
For citation:
G. Pirillo, 2018, "A characterization of Fibonacci numbers" , Chebyshevskii sbornik, vol. 19, no. 2, pp. 259-271.
1. Introduction
Let F0 = 1, F1 = 1 and, for n ^ 2, Fn = Fn-2 + Fn-1 be the Fibonacci numbers. It is well known that = $ = and that in theoretical computer science the Fibonacci
word f = 101101011011010110... is a cutting sequence representing the golden ratio $ (also called Divina Proportione. by Luea Pacioli). Concerning the Fibonacci numbers, the Fibonacci word and the golden ratio, see [3], [7], [14], [9], [10], [11], [12], [13], [15], [16], [17], [18], [19], [20] and [24].
It is also well known that, given three consecutive Fibonacci numbers Fi ^ Fi+1 < Fi+2, the following Cassini identity FiFi+2 — F2+1 = (—1)1 holds. In this paper we support our thesis that the discovery of incommensurability and of the previous equalities came "almost simultaneously", most likely first the Pythagorean identity and immediately after the Cassini identity.
Indeed the Cassini identity is strictly related to the studies and the fundamental results of the Pythagorean School (hereafter simply School) on the incommensurability: side, and diagonal of the. regular pentagon are incommensurable (see Figure 1). The result: if b is the side and a is the diagonal of a regular pentagon, then b : a = a : (b + a) and b(b + a) — a2 = 0 precedes of a very short period of time the discoveries of Fibonacci numbers and Cassini identity FiFi+2 — F2+1 = (—1)* see [19].
2. The irrational number $
The School tried for a long time to find a common measure between the diagonal and the side of the regular pentagon. In the proof of these fundamental results (that we shortly recall hereafter) the following Pythagorean Proposition 1 (see [18]) plays a crucial role (and the same will happen in the first proof of the main result of this paper, Proposition 7).
Proposition 1. Pythagorean Proposition.^) A strictly decreasing sequence, of positive, integers is necessarily finite..
A common measure of diagonal and side of a regular pentagon implies the existence of a segment U and two positive integers ft and a such that U is contained ft times in b, the side, and a time in a, the diagonal. Using elementary results on similar triangles, we easily reach the equalities ft : a = a : (ft + a;) and ft (ft + a) = a2.
But, two such integers ft and a do not exist by an old well-known odd-even argument: i) ft and a both odd implies ft (ft + a) even an d a2 odd (contradictio n), ii) ft odd and a even implies ft (ft + a;) odd an d a2 even (contradiction), iii) ft even and a odd implies ft (ft + a) even an d a2 odd (contradiction), iv) ft and a both even then, using the Pythagorean Proposition 1, we retrieve one of the three previous cases i), ii) and iii) (contradiction). So ft and a cannot be both integers. So side and diagonal of the regular pentagon cannot have a common measure and the following theorem is proved.
в
Рис. 1: Regular pentagon
Теорема 1. Side, and diagonal of the. regular pentagon are. incommensurable..
3. Fibonacci numbers and their relation with incommensurability
We will present hereafter an argument that shows how the Fibonacci numbers and the Cassini identity appeared naturally during the development of the argument of the incommensurability. Several attempts to find a common measure of side and diagonal of the regular pentagon were not successful and will hereafter be examined in depth. Consider two Propositions on the triangle well known today and also well known to the School:
Proposition 2. The. greatest side of a triangle, is that opposite to the. greatest angle..
Proposition 3. The sum of two sides is greater than the. third side..
Considering the isosceles triangle formed by two consecutive sides and by a diagonal of a regular pentagon, the School would have noticed, by Proposition 2, the inequality ft < a and, by Proposition 3, the inequality a < 2ft. This is enough to immediately eliminate the side as a common measure (ft = 1)-
Now, let ft ^ 2. Being ^d a integers, from
ft <a< 2ft,
we have
ft + 1 ^ a ^ 2ft — 1.
Considering the necessary equality ft (ft + a) = a2 and using the above lower bound and upper bound, the School easily eliminated the following segments as common measure: the half of the side (2(2 + 3) — 32 = 0), the third of the side (3(3 + 4) — 42 = 0 and 3(3 + 5) — 52 = 0), the fourth part of the side (4(4 + 5) — 52 = 0 4(4 + 6) — 62 = 0 e 4(4 + 7) — 72 = 0) and so on.
On the other hand, continuing in this way the calculation is increasingly long and difficult as, for each ft > 1, one must consider ft — 1 candidates for a. The departing geometric problem (find a common measure U) is now an arithmetic problem: given an integer ft does there exist an integer a ^ ft such that ft (ft + a) — a2 = 0?
When the recalled argument of incommensurability was completed and consequently it was
ft
considered the just obtained result as a motivation for a new research and has been argumentcd as follows: as ft (ft + a) — a;2 is never 0, we wish to see for what values of ft and a the difference between the greatest and the smallest of the numbers ft (ft + a) and a2 assumes the value 1, which is the minimum possible one. This is a typical curiosity of mathematicians: when they solve a problem, their attention is immediately attracted by the new and often numerous problems that the solution always carries with it. So, we simply believe that, after the discovery of the incommensurability, the School has focused on this new problem.
Today, to find the above recalled values of ft and a is very easy using a computer. It is possible to write a program that searches, finds and puts all these values in the following table. Our brother Mario wrote the program and this is what happens:
Puc. 2: Pythagoreans at work
ft a (X + ft ft (a + ft) a2
1 1 2 12 + 1 12
1 2 3 22 — 1 22
2 3 5 32 + 1 32
3 5 8 52 — 1 52
5 8 13 82 + 1 82
8 13 21 132 — 1 132
13 21 34 212 + 1 212
21 34 55 342 — 1 342
34 55 89 552 + 1 552
55 89 144 892 — 1 892
89 144 233 1442 + 1 1442
144 233 377 2332 — 1 2332
233 377 610 3772 + 1 3772
377 610 987 6102 — 1 6102
610 987 1597 9872 + 1 9872
987 1597 2584 15972 — 1 15972
If, as we think, the School has really tried to find these values of ft and a then they have all noticed the peculiarity of the numbers in the table. The Fibonacci numbers are in the first, second and third column and, in addition, the square of the Fibonacci numbers are in the fifth column while the fourth column contains alternately the predecessor and the successor of these squares, see
[19].
Now, let i ^ 0 and Fj the ith Fibonacci number. Does there exist an integer a ^ Fi such that the difference between the greatest and the smallest of the numbers Fi(Fi + a) and a2 assumes the value 1? Sure, it exists. The table shows that, for each i, 1 ^ Fi ^ 1000, the required number a is exactly Fi+1 and Fi(Fi + Fi+1) — F2+1 = (—1)\ Being Fi + Fi+1 = Fi+2, this equality becomes FiFi+2 — F2+1 = (—1)* and, as it is well-known, the following lemma holds (see for instance [14]).
Lemma 1. Cassini identitity. For each non negative integer i and for each Fibonacci number Fi the following equality holds
FF+2 — F2+i = (—1)\
As we have seen before, step by step the School has picked up new Fibonacci numbers. Each new one discovered corresponded to a more accurate (but not exact!) measurement of the side and diagonal of the regular pentagon. In this sense, the School has discovered and proved the equalitv lim^^ = certainlv not in the verv precise form of the current modern epsilondelta definition that it has todav, but surelv in the sense that the difference $ — became ever smaller and smaller.
4. Cassini identity and characterization of Fibonacci numbers
We introduce a definition which will be crucial in the rest of the paper.
OnPE^EJTEHHE 1. Let, ft a positive integer. When there exists a positive integer a such that, for some non-negative integer 7, the equality
ft (ft + a) — a2 = (—1)t holds, then we say that ft is a Hippasus number and that a is a Hippasus successor of ft.
For the aims of this paper, using the previous definition l, we can obtain a more suitable reformulation of the Cassini identitity.
Proposition 4. If an integer ft ^ 2 is a Hippasus number then any its Hippasus successor a satisfies ft + 1 ^ a ^ 2ft — 1.
Доказательство. . By way of contradiction, suppose a < ft. There exist a positive integer ( such that ft = a + (.Then ft (ft + a) — a2 = (a + ( )(2a + ( ) — a2 = a2 + 3a( + (2 ^ 5. Contradiction. By way of contradiction, suppose a2ft. There exist a positive integer ц > 0, such that a = 2ft + -q. Then ft (ft + 2ft + г() — (2ft + -q)2 = —ft2 — 3ft~q — ri2 ^ —11. Contradiction. Moreover, the equalities
ft(ft + ft ) — (ft )2 = (—1)T ft (ft + 2ft ) — (2ft)2 = (—1)7
are impossible for each integer 7. So if a exist we must have
ft + 1 < a < 2ft — 1.
□
Proposition 4 underlines a clear relation with the geometric origin of the Hippasus number definition.
The proof of the following proposition is very easy.
Proposition 5. For each г ^ 0 the Fibonacci number Fi is a Hippasus number and Fi+i is a Hippasus successor of it.
The following Proposition 6 offers an even more precise reformulation of the Cassini identitity. In order to prove Proposition 7 we need several lemmas.
11
1(1 + 1) — 12 = 1 □
12
1(1 + 2) — 22 = —1 □
1 2 1
Доказательство. For n> 2, we have 1(1 + n) — n2 ^ —5 □
1
Доказательство. Let ft > 1 a Hippasus number and a and a', a = a', both Hippasus successors of ft. By the previous Lemma, we have a > ft and a' > ft.
Without loss of generality, suppose a < Ы. There exists S > 0 and ' non negative integers such that a1 = a + ft (ft + a) — a2 = (—1)7 and ft(ft + a + ô) — (a + ô)2 = (—1)7'. Now,
ft (ft + a + S) — (a + Ô)2 = ft(ft + a) + ftô — (a2 + 2aô + ô2) = (ft(ft + a) — a2) + (ftô — 2aô — ô2 ) = (—1)7 — 5(—ft + 2a + S) = (—1)7 — 5((a — ft)+ a + S). Being a ^ 3 (as a > ft ^ 2), a — ft ^ 1 (as a > ft) and ô ^ 1 (as a' > a), we have (a — ft) + a + S ^ 5 and —5((a — ft) + a + ¿) ^ —5. So
1This terminology seems suitable. Tradition, see [23], attributes to Hippasus the discovery of incommensurability
and our thesis is the following: the discoveries of incommensurability and of a particular class of numbers came
simultaneously, see [19]. So these numbers that we show here to be Fibonacci numbers can provisionally be called
Hippasus numbers.
ft (ft + a + S) - (a + 5)2 < (-1)т - 5 < (—1)т' and ol = a + S cannot be a Hippasus successor of ft. Contradiction. Then two different integers a, a;' cannot be both Hippasus successors of the same ft □
ft
unique Hippasus successor a that is strictly greater than ft. Now, we can precise Proposition 5
Proposition 6. For the Fibonacci numbers the following statements hold:
i) F0 = 1 is an Hippasus number and Fi = 1 is an Hippasus successor of it,
ii) Fi = 1 is an Hippasus numb er and F2 = 2 is an Hippasus successor of it,
Hi) for each г > 1, Fi is an Hippasus numb er and Fi+i is its unique Hippasus successor.
Доказательство, i) follows bv Lemma 2, ii) follows bv Lemma 3 and finally, as for i > 1 we have Fi ^ 2, iii) follows by Proposition 5. □
Lemma 6. Let ft be a Hippasus number and a be a Hippasus successor of ft. Then a — ft ^ ft.
Доказательство. By a trivial verification if ft = 1 and a = 1 and if ft = 1 and a = 2 and by
ft > 1 □
In some sense 0 is a "Hippasus number" having 1 as one of its Hippasus successors (indeed we have 0(0 + 1) — 1 = —1) but by our choice, a Hippasus number must be positive, see Definition 1. For this reason in the next lemma we add the condition a > ft with which we exclude the case ft = 1 and a = 1.
Lemma 7. Let ft be a Hippasus number and a be a Hippasus successor of ft with a> ft. Then a — ft is a Hippasus number and ft is a Hippasus successor of a — ft.
Доказательство. By Lemma 6 we have 0 < a — ft ^ ft. Moreover, we know that for some 7 we have ft(ft + a) — a2 = (—1p. So, (a — ft)((a — ft) + ft) — ft2 = (a — ft)a — ft2 = (—1)[ft(ft + a) — a2] = = (—1)7+i that exactly says that a — ft is a Hippasus number and ft is a Hippasus successor of a — ft □
Lemma 8. Let ft ^ 1 be a Hippasus number and a a Hippasus successor of ft. If a — ft = ft then a — ft = 1 ft = 1 and a = 2.
Доказательство. Consider three cases:
a) ft = 1 a = 1 b) ft = 1 a = 2 and c) ft > 1.
We have a = 2ft Case a): a = 2ft is not true. Case с): a = 2ft is not true by Lemma 4. So, it
□
Now, we are ready to prove the following proposition of which we present two proofs.
Proposition 7. Any Hippasus number is a Fibonacci number.
Доказательство. Let ft be a Hippasus number and let a be a Hippasus successor of it. If ft = 1 and a = 1 then ft is a Fibonacci number. If ft = 1 and a = 2 then ft is a Fibonacci number too.
1
sequence of Fibonacci numbers.)
1ft such a number. We know, by Definition 1, that ft has a Hippasus successor a and, being ft > 1, we also know that a > ft (by Lemma 4) and that a is unique (by Lemma 5).
We know, by Lemma 7, that a — ft is a Hippasus number and that ft is a Hippasus successor of a — ft. By Lemma 6 we have that a — ft ^ ft, i.e., there are two possibilities
a — ft = ft or a — ft < ft.
If a — ft = ft, then by Lemma 8, ft = 1. Contradiction.
So we must have a — ft < ft. Put ft = ft1 and a — ft = ft2.
It may happen that ft1 — ft2 < ft2. Put ft3 = ft1 — ft2.
It may similarly happen that ft2 — ft3 < ft3. Put ft4 = ft2 — ft3.
And so on indefinitely.
In principle, we thus have two possibilities:
-either, for each positive integer k, after the selection of the integer ftk we select ftk+1 with ftk+1 < ftk;
-either the process of selection of ftk+1 strictly smaller of ftk will fail at a certain stage.
Let us take these two possibilities in turn 2.
By Pythagorean Proposition 1 ( an infinite strictly decreasing sequence of positive integers cannot exist) the first possibility cannot happen. So, the process of selection of ftk+1 strictly smaller of ftk will fail at a certain stage when, for a given integer, say i, fti+1 = fti.
So, we suppose that we have selected ft-^, ft2, ..., fti-2, fti-1, fti, fti+1 with a — ft = a — ft1 < ft1, ft1 — ft2 = ft3 < ft2, ft2 — ft3 = ft4 <fts, ..., fti-2 — fti-1 = fti < fti-1 and fti-1 — fti = fti+1 = fti.
By hypothesis ft = ft1 is a Hippasus number and ft2, ..., fti-2, fti-1, fti, fti+1 are all Hippasus numbers bv Lemma 7. Moreover, again bv Lemma 7, fti is a successor of fti+^ fti-1 is a successor of fti, ..., ft1 is a successor of ft2, a is a successor of ft = ft1.
Considering fti-1 — fti = fti+1 = fti, by Lemma 8, we have:
* ftl+1 = 1 = Fo, ftl = 1 = Fh ft- = 2 = F2.
By construction fti-1 = 2 = F2 has a unique Hippasus success or that is fti-2 but, as the Fibonacci number F2 has a unique Hippasus success or that is F3 (see Lemma 6), we have that
ft-2 = 3 = F3.
Similarly,
ft- = 5 = Fa,
fti-4 = 8 = F5,
and, continuing in this way,
ft3 = F-l
ft2 = F.¡-1, fti = Ft.
□
A second proof could be the following. By way of contradiction, suppose that the set of Hippasus numbers which are not Fibonacci numbers is non empty. By the minimum principle this set admits a minimum element, say ft. Necessarily, ft is strictly greater than 2 and has a unique Hippasus successor, say a. Consider a — ft that, by Lemma 7, is a Hippasus number. If a — ft = ft then, by
Lemma 8, ft = 1 that is a Fibonacci number. Contradiction. If a — ft < ft then, by Lemma 1, a — ft
ft
The second proof, that uses the minimum principle, is shorter than the first one, which we prefer as it uses explicitly the Pythagorean Proposition 1.
Proposition 5 and Proposition 7 imply the following
Proposition 8. A positive integer is a Hippasus number if, and only if, it is a Fibonacci number.
2Here we try to imitate a clear, elegant and powerful model of exposition that Ramsey presented in [22].
By our previous results we are convinced that the relations between the Pythagorean identity b(b+a) — a2 = 0 and the Cassini Identity ft (ft + a) —a2 = (—1)7 are really very strict. At least in our thesis, the School, that discovered the first equality, hardly could have ignored the second one. In other terms, when the School found a Hippasus number then the same School simultaneuslv found a Fibonacci number, because no other number could have be found. In order to add another argument to our previous ones (in particular Proposition 4), we prove directly the following proposition.
Proposition 9. Let ft be a Hippasus number and a be a Hippasus successor of ft. Then a is a Hippasus number and a + ft is a Hippasus successor of a.
Доказательство. For some 7 we have (—1)7 = ft(ft + a) — a2. So a(a + a + ft) — (a + ft)2 = = a2 + a(a + ft) — (a + ft)2 = a2 — aft — ft2 = (—1)(—a2 + aft+ft2) = (—1)(ft (ft + a) — a2) = (—1)7+1 i.e. a is a Hippasus number and a + ft is a Hippasus successor of it. □
Следствие 1. If a is a Hippasus number and ft is its Hippasus successor then a + ft is a Hippasus number.
Corollary 1 certifies that the laws of formation of Fibonacci numbers and of Hippasus numbers are the same! Much better, the Fibonacci law Fn + Fn+i = Fn+2 rediscovers the Pythagorean law given in the previous Corollary 1. Moreover, the Definition 1 of Hippasus numbers is operational and allows us to find Hippasus numbers one after the other.
The Wasteel result of next section is just a criterion to decide if two integers are consecutive Fibonacci numbers.
5. With Fibonacci numbers the surprises never end
Dickson recalls in [8] the following result of Wasteels, proved in [24].
Proposition 10. Two positive integers x and у for which y2 — xy — x2 equals +1 or —1 are consecutive terms of the series of Fibonacci.
Mativasevich in [13] with reference to the result of Wasteels says: The fact that successive Fibonacci numbers give the solution of Eq. (25) was presented by Jean-Dominique Cassini to the Academie Roy ale des Sciences as long ago as 1680. It can be proved by a trivial induction. At the same time the stronger fact that Eq. (25) is characteristic of the Fibonacci numbers is somehow not given in standard textbooks. The induction required to prove the converse is less obvious, and that fact seems to be the reason for the inclusion of the problem of inverting Cassini's identity as Exercise 6.44 in Concrete Mathematics by Ronald Graham, Donald Knuth, and Oren Patashnik [13]. As the original source of this problem the authors cite my paper [21], but I have always suspected that such a simple and fundamental fact must have been discovered long before me. This suspicion turned out to be justified: I have recently found a paper of M. Wasteels [4-1] published in 1902 in the obscure journal Mathesis. 3
A pentagon on a portale of "Duomo di Prato" refers to Fibonacci numbers 4 and a octagon on the same portale seems to have a reference to a singular construction of an octagon that uses Fibonacci numbers! This octagon is not regular but very impressively similar to a regular octagon: we design two concentric circles having diameters Fn and Fn+2, the two horizontal straight line tangent to the inner circle and the two vertical straight line tangent to the same inner circle. These four lines
3In this citation Eq. (25) is the Cassini identity FiFi+2 — F?+1 = (—1)\ Paper [13] corresponds to [11] here, paper [21] is the fundamental and historical paper of Matiyasevich (here [12]) and paper of Wasteels [41] is [24] here.
4Recently, the Fibonacci numbers have been rediscovered in a tarsia of the Church of San Nicola in Pisa (see
Armienti [2] and Albano [1]).
cut the larger circle into 8 points. We denote by Pn and Qn the two of them having the following coordinates and lying in the first quadrant:
P = I ^
rn = \ 2 '
(f. v(M2—M2) ■= G/(v)2—• *)
IS
They are the extremes of one of the eigth sides of our octagon. We note that their distance dn V2 y^^2) — ^^r j — ^f . We also denote bv en the side of the regular octagon inscribed in
the circle of diameter Fn+2. We have that:
-the value f^ tends to the limit ^ — 1 — 1
i.e. about 1.00375,
the value ^ tends to the limit V1 — $-4 — $-2
the value tends to the limit
ri
\/2-— V2-V2 $2
, i.e. about 1.001874,
, i.e. about 1.001878.
It seems that the architech of the "Duomo di Prato'' was Carboncettus marmorarius see [5] and [6]. For these reasons one can speak about Carboncettus octagon!
Puc. 3: The portal of the Duomo of Prato
6. Conclusions
In this paper we reconsider two of our old questions: when, for the first time, the Fibonacci Numbers were mathematically well defined and who defined them? Conventional wisdom suggests that the Fibonacci Numbers were first introduced in 1202 by Leonardo of Pisa, better known today as Fibonacci, in his book Liber abbaci. The intent of this article is to offer a plausible conjecture on the origin of the Fibonacci Numbers. Indeed, our paper contains comments on the relationship between golden ratio and the Fibonacci Numbers. We try to imagine the work of the Pythagorean School and the first steps that led to their discovery of the irrational number the golden ratio. We suppose that before discovering that no common measure was possible for the side and diagonal of a regular pentagon, in particular they verified that: i) the side was not a common measure, ii) the half of the side was not a common measure, iii) the third of the side was not a common measure and so on. We analyze these "unsuccessful" attempts, during this analysis we realize that the Fibonacci Numbers appear and we conclude that probably the Pythagorean School also noticed ... these same Fibonacci Numbers! Finally we would like to point out that this paper is solely based on some remarks about the arguments used by the Pythagorean School and not on historical documents.
In conclusion, during our personal investigation on the discovery of the irrational numbers (made in the absolute absence of documents), we find enough traces of the Fibonacci numbers and
of their properties to convince us that these numbers were born in Crotone in the VI-IV century B.C. On the other hand, stricto sensu we present no historical discovery but we present a new characterization of the Fibonacci numbers and, perhaps, we provide maths teachers some useful educational suggestion.
7. Acknowledgements
I thank my brother Mario, lecturer at the Educandato Santissima Annunziata in Firenze, for the program that permitted us to build the table of this article. I thank also Maurizio Aristodemo, Luigi Barletti, Gabriele Bianchi, Marco Pellegrini, Carlo Toffalori, Gabriele Villari, for a first reading of this paper. I thank the Dipartimento di Matematica e Informatica "Ulisse Dini" for his hospitality.
СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ
1. Albano, A. The Fibonacci sequence and the golden section in a lunette decoration of the medieval church of San Nicola in Pisa // Territori della Cultura, 21 ottobre 2015 (printed in 2016), pp. 48-59.
2. Armienti, P. The medieval roots of modern scientific thought. A Fibonacci abacus on the facade of the church of San Nicola in Pisa // Journal of Cultural Heritage, 2016, vol. 17, pp. 1-6.
3. Arnoux, P., Siegel, A. Dynamique du nombre d'or // Actes de l'université d'été de Bordeaux, http://iml.univ-mrs.fr/~arnoux/articles.htiiil. 2004, In press.
4. Alfred, Brother U. An Introduction to Fibonacci Discovery / San Jose, Calif., Fibonacci Association, San Jose State College, 1965.
5. Cerretelli, C. , L'architettura della chiesa //II Duomo di Prato, Casa Editrice Le Lettere (Cariprato), 2009, Firenze, pp. 57-145.
6. Fantappiè, R. Le carte della propositura di S. Stefano di Prato / I, 1006-1200, Firenze, Leo S. Olschki Editore, 1977, pp. 332-335.
7. Fibonacci, [Leonardo Pisano, Bigollo], Liber abbaci / Scritti di Leonardo Pisano matematico del secolo decimoterzo, 1857, vol. 1: II Liber abbaci di Leonardo Pisano / pubblicato secondo la lezione del Codice Magliabechiano C. 1., 2616, Badia Fiorentina, n. 73 da Baldassarre Boncompagni, socio ordinario dell'Accademia pontificia de' nuovi Lincei, Roma, Tipografia delle scienze matematiche e fisiche.
8. Dickson, L. E. History of the Theory of Numbers / vol. 1, Divisibility and primalitv, Washington, 1919, Carnegie Institution of Washington.
9. Knuth, D.E. The Art of Computer Programming / vol. 1: Fundamental Algorithms, Reading, Mass., 1968, Addison-Wesley.
10. Knuth, D.E., Morris, J.H., Pratt, V. R., Jr., Fast pattern matching in strings // SIAM J. Comput., 1977, vol. 6, no. 2, pp. 323-350.
11. Graham, R. L., Knuth, D. E., Patashnik, O. Concrete Mathematics: a foundation for computer science / Reading, Addison-Wesley, 1999.
12. Mativasevich, Yu. V. Enumerable sets are Diophantine // Soviet. Math. Dokladv, 1970, vol. 11, no. 2, pp. 354-358.
13. Matiyasevich, Yu. V. Hilbert's Tenth Problem: Diophantine Equations in the Twentieth Century / (translated by R. Cooke), in "Mathematical Events of the Twentieth Century" (edited by A. A. Bolibruch, Yu. S. Osipov, and Ya. G. Sinai), Springer, Berlin, 2003, 185-213.
14. Meyer, A., Stevaert, C. Le nombre d'or et les nombres de Fibonacci / IREM [Institut de recherche sur l'enseignement des mathématiques], Université Paris VII, Paris, 1981.
15. Pirillo, G. Fibonacci numbers and words, // Discrete Math., 1997, vol. 173, no. 1-3, pp. 197-207.
16. Pirillo, G. A curious characteristic property of standard Sturmian words // In: Algebraic combinatorics and computer science, 2001, Springer Italia, Milan, pp. 541-546.
17. Pirillo, G. Inequalities characterizing standard Sturmian and episturmian words // Theoret. Comput. Sci., 2005, vol. 341, no. 1-3, 276-292.
18. Pirillo, G. Numeri irrazionali e segmenti incommensurabili // Nuova Secondaria, 2005, vol. 7, 87-91.
19. Pirillo, G. Some recent results of Fibonacci numbers, Fibonacci words and Sturmian words // IASI Research Report n. 16-07, Southeast Asian Bull, of Math. 2016, To appear.
20. Pirillo, G. La scuola pitagorica ed i numeri di Fibonacci // Archimede, 2017, vol. 2, 66-71.
21. Pirillo, G. Figure geometriche su un portale del Duomo di Prato // Prato Storia e Arte, 2017, 121, 7-16.
22. Ramsey, F.P. On a problem of formal logic // Proc. London Math. Soc., 1929, vol. 30, no. 4, 264-286.
23. von Fritz, K. The discovery of incommensurability by Hippasus of Metapontum // Ann. of Math., 1945, second series, vol. 46, pp. 242-264.
24. Wasteels, M. J. Quelques Propriétés des Nombres de Fibonacci // troisième sèr., Mathesis, 1902, vol. 3, pp. 60-62.
REFERENCES
1. Albano, A. 2016, "The Fibonacci sequence and the golden section in a lunette decoration of the medieval church of San Nicola in Pisa", Territori della Cultura, 21 ottobre 2015 (printed in 2016), pp. 48-59.
2. Armienti, P. 2016, "The medieval roots of modern scientific thought. A Fibonacci abacus on the facade of the church of San Nicola in Pisa", Journal of Cultural Heritage, vol. 17, pp. 1-6.
3. Arnoux, P., Siegel, A. 2004, "Dynamique du nombre d'or", Actes de l'université d'été de Bordeaux, http://iml.univ-mrs.fr/~arnoux/articles.html. In press.
4. Alfred, Brother U. 1965, An Introduction to Fibonacci Discovery, San Jose, Calif., Fibonacci Association, San Jose State College.
5. Cerretelli, C. 2009, "L'architettura della chiesa" in: II Duomo di Prato, Casa Editrice Le Lettere (Cariprato), Firenze, pp. 57-145.
6. Fantappiè, R. 1977, Le carte della propositura di S. Stefano di Prato, I, 1006-1200, Firenze, Leo S. Olschki Editore, pp. 332-335.
7. Fibonacci, [Leonardo Pisano, Bigollo], 1857, "Liber abbaci", Scritti di Leonardo Pisano matematico del secolo decimoterzo, vol. 1: II Liber abbaci di Leonardo Pisano / pubblicato secondo la lezione del Codice Magliabechiano C. 1., 2616, Badia Fiorentina, n. 73 da Baldassarre Boncompagni, socio ordinario dell'Accademia pontificia de' nuovi Lincei, Roma, Tipografia delle scienze matematiche e fisiche.
8. Dickson, L.E. 1919, History of the Theory of Numbers, vol. 1, Divisibility and primalitv, Washington, Carnegie Institution of Washington.
9. Knuth, D.E. 1968, The Art of Computer Programming, vol. 1: Fundamental Algorithms, Reading, Mass., Addison-Wesley.
10. Knuth, D. E., Morris, J. H., Pratt, V. R., Jr., 1977, "Fast pattern matching in strings", SIAM J. Com,put., vol. 6, no. 2, pp. 323-350.
11. Graham, R. L., Knuth, D.E., Patashnik, O. 1999, Concrete Mathematics: a foundation for computer science, Reading, Addison-Wesley.
12. Mativasevich, Yu. V. 1970, "Enumerable sets are Diophantine", Soviet. Math. Doklady, vol. 11, no. 2, pp. 354-358.
13. Mativasevich, Yu.V. 2003, Hilbert's Tenth Problem,: Diophantine Equations in the Twentieth Century (translated by R. Cooke), in "Mathematical Events of the Twentieth Century" (edited by A. A. Bolibruch, Yu. S. Osipov, and Ya. G. Sinai), Springer, Berlin, 185-213.
14. Meyer, A., Stevaert, C. 1981, Le nombre d'or et les nombres de Fibonacci, IREM [Institut de recherche sur l'enseignement des mathématiques], Université Paris VII, Paris.
15. Pirillo, G. 1997, "Fibonacci numbers and words", Discrete Math., vol. 173, no. 1-3, pp. 197-207.
16. Pirillo, G. 2001, "A curious characteristic property of standard Sturmian words". In: Algebraic combinatorics and computer science, Springer Italia, Milan, pp. 541-546.
17. Pirillo, G. 2005, "Inequalities characterizing standard Sturmian and episturmian words", Theoret. Comput. Sci., vol. 341, no. 1-3, 276-292.
18. Pirillo, G. 2005, "Numeri irrazionali e segmenti incommensurabili", Nuova Secondaria, vol. 7, 87-91.
19. Pirillo, G. 2016, "Some recent results of Fibonacci numbers, Fibonacci words and Sturmian words", IASI Research Report n. 16-07, Southeast Asian Bull, of Math. To appear.
20. Pirillo, G. 2017, "La scuola pitagorica ed i numeri di Fibonacci", Archimede, vol. 2, 66-71.
21. Pirillo, G. 2017, "Figure geometriche su un portale del Duomo di Prato", Prato Storia e Arte, 121, 7-16.
22. Ramsey, F.P. 1929, "On a problem of formal logic", Proc. London Math. Soc., vol. 30, no. 4, 264-286.
23. von Fritz, K. 1945, "The discovery of incommensurability by Hippasus of Metapontum", Ann. of Math., second series, vol. 46, pp. 242-264.
24. Wasteels, M.J. 1902, "Quelques Propriétés des Nombres de Fibonacci", troisième sèr., Mathesis, vol. 3, pp. 60-62.
Получено 11.06.2018 Принято в печать 17.08.2018
