REPRESENTATION OF REAL NUMBERS BY MEANS OF FIBONACCI NUMBERS

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1 REPRESENTATION OF REAL NUMBERS BY MEANS OF FIBONACCI NUMBERS Autor(en): Objekttyp: Ribenboim, Paulo Article Zeitschrift: L'Enseignement Mathématique Band (Jahr): 31 (1985) Heft 1-2: L'ENSEIGNEMENT MATHÉMATIQUE PDF erstellt am: Persistenter Link: Nutzungsbedingungen Die ETH-Bibliothek ist Anbieterin der digitalisierten Zeitschriften. Sie besitzt keine Urheberrechte an den Inhalten der Zeitschriften. Die Rechte liegen in der Regel bei den Herausgebern. Die auf der Plattform e-periodica veröffentlichten Dokumente stehen für nicht-kommerzielle Zwecke in Lehre und Forschung sowie für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und den korrekten Herkunftsbezeichnungen weitergegeben werden. Das Veröffentlichen von Bildern in Print- und Online-Publikationen ist nur mit vorheriger Genehmigung der Rechteinhaber erlaubt. Die systematische Speicherung von Teilen des elektronischen Angebots auf anderen Servern bedarf ebenfalls des schriftlichen Einverständnisses der Rechteinhaber. Haftungsausschluss Alle Angaben erfolgen ohne Gewähr für Vollständigkeit oder Richtigkeit. Es wird keine Haftung übernommen für Schäden durch die Verwendung von Informationen aus diesem Online-Angebot oder durch das Fehlen von Informationen. Dies gilt auch für Inhalte Dritter, die über dieses Angebot zugänglich sind. Ein Dienst der ETH-Bibliothek ETH Zürich, Rämistrasse 101, 8092 Zürich, Schweiz,

2 2) Every x, 0 < x < S is representable by the séquence (s). i 1 REPRESENTATION OF REAL NUMBERS BY MEANS OF FIBONACCI NUMBERS by Paulo Ribenboim Dedicated to Professor D.H. Lehmer at the occasion ofhis eightieth birthday The purpose of this note is to dérive a new représentation of positive real numbers as sums of séries involving Fibonacci numbers. This will be an easy application of an old resuit of Kakeya [4]. The paper concludes with a resuit of Landau [5], relating the sum ]T? with values of thêta séries; we believe it worthwhile to unearth Landau's resuit, which is now rather inaccessible. l 1. Let (s / ).^ 1 be a séquence of positive real numbers, such that s { > s 2 > s 3 >... and lim s t? 0. Let S = s t. i-><x) i= 1 We say that x > 0 is representable by the séquence (s,)^ if x = s t. (with i 1 < i 2 < i 3 <...). Then necessarily x S. The first resuit is due to Kakeya ; for the sake of completeness, we give a prf : Proposition 1. The following conditions are équivalent: 1) Every x, 0 < x S, is representable by the séquence (s t ),x = Y, s t?> j= i w/zere z^ is t/ie smallest index such thaï s h < x. 3) For every n 1, sn s t. i = n+ 1 Prf. 1 -> 2. This is trivial. 2 -> 3. If there exists n 1 such that s n > s i9 let x be such that i = n+ 1 s n > x > Y s i- fi y hypothesis, x = st. with ï x < i 2 <... Since i = n+l j=i s«> x > 5 fi, then n < i l9 hence x -. < 5fc, which is absurd. j~l k=n+l

3 3 -* 1. Since lim s t = 0, there exists the smallest index i 1 such that s h < x. i~+ Similarly, there exists the smallest index i 2 such that i 1 < i 2 and s h < x? s ir More generally, for every O 1 we define i n to be the smallest index n-l such that i n _ 1 < i n and s in < x? Yj s ij- Then x s tj. Suppose that x > s f. 7=1 ' 7=1 m We note that there exists N such that if m N then s.- < x? Y s }. Otherwise, there exist infinitely many indices n 1 < n 2 < n 3 <... nu. 7=1 such that s x - Y S;.. At the lirait, we hâve 0 = lim s ; x? Y s, > 0, and K 7=1 /c-> k 7=1 this is a contradiction. We chse N minimal with above property. We show : for every m N, im + 1 = i m + 1. In fact so by définition of the séquence of indices, i m + 1 = i m + l. Therefore the following sets coincide: {i N, i N +l, i N + 2,...} = {i N, i N+l9 i N + 2,...}. Next we show that i N = 1. If i N > 1 we consider the index i N? 1, and by hypothesis (3) : We hâve i N _ 1 i N? 1 < i N.lfiN - 1 < i N? 1 this is impossible, because i N was defined to be the smallest index such that i N _ 1 < i^ and s in < x N-l N-l? ]T s f.. Thus i^.! = i N? 1, that is s in _ l < x? s f. and the choice of Af as minimal with the property indicated. this is against Thus i N = 1 and x > s tj = ]T 5^ = S, against the hypothesis.? We remark now that if the above conditions are satisfied for the séquence (s i ) i>1, if m 0 then every x, 0 < x < Sf = s t is representable by i = m+ 1 the séquence (Sj) f>fn+1 with i 1 the smallest index such that m + 1 ï'i and s ix < x.

4 Indeed, condition (3) holds for (s I ). >1 hence also for (si) - i>m + l Since 0 < x < S', the remark follows from the proposition. Proposition 1 has been generalized (see for example Fridy [3]). Now we consider the question of unique représentation (this was generalized by Brown in [1]). Proposition 2. With above notations, the following conditions are équivalent : 2') Every x,0<x<s, has a unique représentation x = s f.. 3') For every n 1, s n = s t? i = n + 1 4') For every n 1, sn =?7 s x (hence S = 2s 1 ). Prf. 2' -> 3 f. Suppose there exists n 1 such that s n s t. i = n+l Since (2') implies (2) hence also (3) then s n < s f Let x be such that i = n+l s n < x < J] s t? i-n + l Sy the above remark, x is representable by the séquence {sj.^ +i, that is x = Sfcr On the other hand, (2') implies (2), hence fc j Z n + 1 also (1) and x has a représentation x = s. where is the smallest index such that s tl < x. From s n < x it follows that i 1 < n and so x would hâve two distinct représentations, against the hypothesis. 3' -+ 4'. We hâve s n = s n * = 2s n + 1 for every w 1 hence s n = i^î s i for evefy w > i- 4' -» 2'. Suppose that there exists x, 0 < x < S having two distinct représentations Let ; 0 be the smallest index such that i jo k jo, say i jo < k jo. Then.»., = z *, < i *..

5 By hypothesis, after dividing by s 1, we hâve hence 2 1 ~ lj 0, which is impossible. j = jo+l For practical applications, we note : If s n 2s n+l for every n 1 then condition (3) is satisfied. Indeed and s n n + 1 si i= n Now we give various ways of representing real numbers. First, the dyadic représentation, which may of course be easily obtained directly : Corollary 1. Every real number x,0<x< 1, may intheform x =? - (with I^n 1 <n 2 <n 3 <...). j=i2 J be written uniquely Prf. This has been shown in proposition 2, taking s x = -.?, Corollary 2. Every positive real number x may x = \ Y? (with n 1 <n 2 <n 3 <...). be written in the form Prf. We consider the séquence ( - ), which is decreasing with limit \ njn>l 1 12 equal to zéro, and we note that Y - = and - for every n 1.? = i n n n + 1 Thus by Kakeya's theorem and the above remark, every x > 0 is represent ableas indicated. D [

6 Corollary 3. Every positive real number x may be written as» i x =? j=ipij (where p <p 1 2 <P3<--- numbers). * 5 the increasing séquence of prime Prf. We consider the séquence (? 1, which is decreasing with l limit equal to zéro. As Euler proved Y? =. By Tschebycheffs theorem i=l Pi (prf of Bertrand's "postulate") there is a prime in each interval (n, 2ri) ; 1 2 thus p i + 1 < 2 pi and? < for every i 1. By Kakeya's theorem and Pi Pi + i the above remark, every x > 0 is representable as indicated.? 3. Now we shall represent real numbers by means of Fibonacci numbers and we begin giving some properties of thèse numbers. The Fibonacci numbers are: F 1 = F 2 = 1 and for every n 3, F n is defined by the récurrence relation F n = F n _ 1 + F n _ 2. Thus the séquence oï Fibonacci numbers is In the following proposition, we give a closed form expression for the Fibonacci numbers; this is due to Binet (1843). Let a =? (the golden number) and P = -?, so a + p = 1, ap = - 1, thus a, p are the rts of X 2 -l<p<o<l<ot. We hâve - X - 1 = 0 and Lemma 1. For every n 1, F n = -J- and < F < x/5./5./5 a"? 6" Prf. We consider the séquence of numbers G n =? for n 1 Then G x = G 2 = 1 ; moreover

7 because a 2 = a H- 1, p 2 = P + 1. Therefore the séquence (G n ) n>1 coincides with the Fibonacci séquence. Now we establish the estimâtes. If n > 1 then (-p)" = \ < oc"" 1 = - a"p = a"(a-l) = a" a", so Similarly, if n 2 then (-(3)" =? < a"' 2 = -a""^ = «""^a-l) a " Q" a «/ Q\n - 1 = a"? a"" 1 so F n =??? >?? n = i.. V5 75 ; this is also true when n For every m > 1 let I m =?^ We hâve : Lemma 2. For every m 1, 7 m <, J x < J 2 < 7 3,..., and lim I m = go. m-> Prf. We hâve noting that -^ < 1 Next, we hâve Finally thus lim I m m-* =.

8 Proposition 3. For every positive real number x there exists a unique J m > 1 such thaï x = -ï^ fcwt 9 x zs no^ of the form gi/on-ir Prf. First we note that each of the séquences (?^ J is decreasing with limit equal to zéro. By the above proposition, there exists m 1 such that I m _ 1 < x I m (with 7 0 = 0) m. We observe that? for m 1, because F n + 1 = F n Fn Fn+1 F n+1 + i7^! < 2 F n. By proposition 1 and a previous remark, x is representable l as indicated, while the last assertion follows from x > I m _ 1 =? 1/m-1. i = 1 -T ; l The number / A =? appears to be quite mysterious. As we hâve n = i F n seen V5 < 7 t <. a? 1 4. In 1899, Landau gave an expression of I 1 in terms of Lambert séries and Jacobi thêta séries. The Lambert séries is L(x) = -; it is convergent for 0 < x < 1, as easily verified by the ratio test. Jacobi thêta séries, which are of crucial importance, for example in the theory of elliptic functions, are defined as follows, for 0 < q \ < 1 and zgc\ OO -v-"

9 In particular, we hâve Now we prove Landau's resuit : Proposition 4. We hâve : 1) 2) Prf: 1 v^ 75 y^p" -pr-p so Since p < 1, then

10 2) Now we hâve Now we need to détermine the coefficient of p m (for m odd) remarking that since the séries is absolutely convergent, its terms may be rearranged. If m is odd and d divides m, then (3 m appears in the horizontal Une m beginning with? p 7 with the sign Thus the coefficient e m of (3 m is em = 5 3(m) - ô^m) where A well-known resuit of Jacobi (see Hardy & Wright's bk, page 241) relates the différence ô^m)? ô 3(m) with the number r(m) = r 2 (m) of repré sentationsof m as sums of two squares. Precisely, let r(m) dénote the number of pairs (s, t) of integers (including the zéro and négative integers) such that m = s 2 + t 2. Jacobi showed that It follows that the number r'(m) of pairs (s, t) of integers with s > t 0 and m = s 2 + t 2 is (the first summand above corresponds to the représentation of m as a sum of two non-zero squares).

11 Therefore Since m is odd then s t (mod 2) and therefore So We may now express this formula in terms of Jacobi séries. Namely so An unpublished formula of Gert Almqvist (1983) gives another expression of 7\ only in terms of Jacobi thêta séries : Carlitz considered also in 1971 the foliowing numbers 1»=1 t n

12 Clearly, ail the above séries are convergent. Carlitz showed that S 3,S 7,S u,...eq(v / 5), while S«= r k + r' k S 0 for k> 1 and r k,r'k eq. One may ask: what kind of number is S o l Are the numbers S 0,S l9 S 2 algebraically independent? BIBLIOGRAPHY [1] Brown, J. L. On generalized bases for real numbers. Fibonacci Q. 9 (1971), [2] Carlitz, L. Réduction formulas for Fibonacci summations. Fibonacci Q. 9 (1971), and 510. [3] Fridy, J. A. Generalized bases for the real numbers. Fibonacci Q. 4 (1966), [4] Kakeya, S. On the partial sums of an infinité séries. Science Reports Tôhoku Imp. Univ. (1), 3 (1914), [5] Landau, E. Sur la série des inverses de nombres de Fibonacci. Bull. Soc. Math. France 27 (1899), [6] Sansone, G. and J. Gerretsen. Lectures on the Theory of Functions of a Complex Variable, Vol. I. P. Nrdhoff, Groningen, (Reçu le 22 octobre 1984) Paulo Ribenboim Department of Mathematics and Statistics Queen's University Kingston, Ontario Canada

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