(2) F j+2 = F J+1 + F., F ^ = 0, F Q = 1 (j = -1, 0, 1, 2, ). Editorial notes This is not our standard Fibonacci sequence.

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1 ON THE LENGTH OF THE EUCLIDEAN ALGORITHM E. P. IV1ERKES ad DAVSD MEYERS Uiversity of Ciciati, Ciciati, Ohio Throughout this article let a ad b be itegers, a > b >. The Euclidea algorithm geerates fiite sequeces of oegative itegers, such that (1) {q} 1 1 ad J J = l a = qtb + r if b = q 2 ri + r 2, * i = Q3 r 2 + r 3, < < < ur 3 3= 1 ri < r 2 < r 3 < b, * i «^2. r - 3 = q - l r r - l ' r = q r + r, -2-1 < r < -1 r = r - 2 The itegers r _- is the greatest commo divisor of a ad b ad q ^ 2. Defie I(a, b) to be the umber of divisios i the algorithm (1). Some basic properties of i(a,b) are (i) i(a,a) = 1 ; (ii) i(ac,bc) = i(a,b), c > ; (iii) i(a + b,b) = i(a,b) ; (iv) I (a + b,a) = 1 + i(a,b). Each of these properties is proved directly from the defiitio (1). Property (ii) permits us to assume a ad b are relatively prime. This paper is cocered with maximizig i(a,b) whe the itegers a ad b are draw from certai subclasses of positive itegers. There are some classical results i this directio such as the theorem of Lame [3, p. 43] which states that i(a,b) tha five times the umber of digits i b. is ever greater We begi with a kow result, the proof of which is istrumetal for the justificatio of the mai theorem of the paper. Theorem 1. Let { F. } be the Fiboacci sequece geerated by (2) F j+2 = F J+1 + F., F ^ =, F Q = 1 (j = -1,, 1, 2, ). Editorial otes This is ot our stadard Fiboacci sequece. 56

2 F eb ON THE LENGTH OF THE EUCLIDEAN ALGORITHM 57 If a < F m + 1 or b < F m for some iteger m >, the i(a,b) < i(f + 1, F ) = m. Proof. From (1) the ratioal umber a/b has a cotiued fractio expasio (3) B - * < *, &*!<">. ^ 2. The k umerator A, ad the k deomiator B. of this cotiued fractio a r e determied from the equatios ( 4 ) A k = \ A k - l + A k - 2 ' B k = \ B k-l + B k-2 * = 1, 2,, ), where A = 1, B =, A 4 = q ls Bt = 1 [2, p. 3 ]. Sice q, > for each idex k ^ 5 it follows from (4) that \ > A k - r B k > B k - i * t ). Moreover, by J (1) ad (4) we have a ^ A, b ^ B. Suppose a ad b are itegers for which = i(a,b) m. Sice q, we have A = F, Aj ^ Fi, A 2 ^ Ft + F = F 2, ad, i geeral, 1 (1 ^ k ^ ), A k A k A k - 2 * F k F k - 2 = F k (1 < k < ). Fially, sice q ^ 2, we have by (2) A > 2 A - + A ^ 2F - + F = F - + F = F _ L l Similarly, B, ^ F, - (1 ^ k < ) ad B ^ F. Furthermore, A = F,- if ad oly if J k k-1 +1 a = 1 (1 ^ k < ), q = 2 ad B = F if ad oly if q, = 1 (1 < k < ), q = 2. Sice a ^ A ^ F,. - F, - ad b ^ B ^ F ^ F, we have the cotrapositive of the +1 m+1 m first part of the implicatio i the statemet of the theorem proved, The fact that i(f - F ) = m is a cosequece of the statemets cocerig equality of A ad B with F - ad F, respectively [ l ]. if a ^ a? The ordered pairs of itegers (a,b) ca be partially ordered by defiig (a, b)#(a T,b f ) ad b ^ b f. Relative to this partial order, the theorem states, i particular, that (F + 1, F ) is the "first" pair for which I(a,b) = m, i. e,, if (a!, b')»(f + 1, F ), the i ( a!, b f ) < i(f, -, F ) uless a? = F,- ad b f = F. m+1 m m+1 m The proofs of our ext results are depedet o the followig kow lemma. Lemma 1. F, = F F + F. F - (p,q = 1,2,- ). P +c 3 P q p-1 q-1 Proof. Set S = F F + F, F -. The by (2) p,q p q p-1 q-1 S = ( F + F ) F + F F = F F + F F = S p,q p-1 p-2 q p-1 q-1 p-1 q+1 p-2 q p - l, q + l

3 58 ON THE LENGTH OF THE EUCLIDEAN ALGORITHM [Feb. Repeated applicatio of this idetity yields S = S = F F + F F = F p,q l,q+p-l 1 p+q-1 p+q-2 p+q Corollary (Lame). If m is the umber of digits i the iteger b, the i(a,b) ^ 5m. Proof. We first show F > 1 by iductio. For = 1, F 6 = 13 > 1, If the iequality is valid for a iteger, the by Lemma 1 sice F* ^ = F^ ^ F. + F r F. > I 1 = ^ 1 > l 5 b TT > I r r 5 2 r 5+l ' Thusj the iequality is valid for all itegers. Now if b has m digits, the b < 1 ad, hece, b < F g m + 1 > By Theorem 1 it follows that i(a,b) < 5m + 1 ad Lame theorem is proved. It is iterestig to observe that equality is possible i Lame theorem if b < 1 3. If b has four digits, the b < F 2 = 1946 ad, by Theorem 1, i(a,b) < i(f 2 1,F 2 ) = 2. More geerally, equality caot hold i the Corollary for m > 3. Ideed, by Lemma 1 ad the argumet used i the proof of the corollary, we have F > 1 implies F _ > 1 Sice F 2 > 1 4, it follows that F 5 m > 1 m for m ^ 4. If b < 1 m (m ^ 4), the i(a,b) s < i ( F r x 1, F c ) = 5m. 5m+l 5m The ext problem cosidered i this article pertais to the umber of distict pairs (a, b) such that ad i(a,b) = m. (F _,_-, F )a(a f b)qf(f ^ F ^ ) m+1 m m+2 m+1 We prove there are m + 1 such pairs ad obtai formulas for the itegers a ad b that comprise the pairs. It is coveiet to establish these results from a sequece of lemmas. Lemma 2. Let the Euclidea algorithm for a ad b, a ad b are relatively prime, be (1) where for some iteger m (1 < m < ) - q = 2 ad q, = 1 (k f m, 1 ^ k < ), q = 2. The M ad a ~ F +1 + F - m + l F m - l b = F + F _ F Q. -m+1 m-2 Moreover, (a,b)<*(f. 9> F. - ).

4 1973] ON THE LENGTH OF THE EUCLIDEAN ALGORITHM 59 Proof. From the proof of Theorem 1, we have that the k umerator ad deomiator of the cotiued fractio expasio for a/b whe i(a,b) = satisfy, for k < m, the coditios A k = F k > B k = F k _ r From this fact ad (4), we have A = 2 F - + F = F + F - = F + F f t F -, m m - 1 m-2 m m - 1 m m - 1 B m = 2 F m F m - 3 = F m F m - 2 = F m F F m - 2 A m+1 = ( F m + ^ - l * + F m - 1 = F m F l F m - l > B - L l = ( Thus, by iductio, we obtai F i + F o ) + F o = F + F i F o m+1 m - 1 m-2 m-2 m 1 m-2 Fially, by (4) ad these formulas, A 1 = F + F F -, -1-1 m m - 1 B - = F + F F m-2 - m - 1 A = 2F, + F + (2F _ + F ) F - = F _,, + F _ F m+1 -m-2 m m+1 m - 1 ad, similarly, J B = F + F,-F. Therefore, a = A ad b = B ad the first ' -m+1 m-2 part of the lemma is proved. Next, by Lemma 1, it follows that F, < A = F _ L l + F ^ F 1 = F _ L 1 + F - F F < F _! _ m+1 m ^m m-2 +2 ad, similarly, F R < B R < F R+r This lemma gives us - 2 pairs (m = 2, 3, *, - 1) of itegers (a,b) such that F ^ < a < F _^, F < b < F ^, +1 +2? +1 ad i(a,b) =. Sice ^ ( F + 1 * F ) ad i ( F + 2> F > = i ( F + l + F > V = i ( F + l > V = > there are so far pairs i the rage (F _,,, F )a(a,b)a(f ^, F _,, ) +l J ' +2' +1 for which i(a,b) =. the ext two lemmas. The fact that there exists oly oe additioal such pair is proved by

5 6 ON THE LENGTH OF THE EUCLIDEAN ALGORITHM [Feb. Lemma 3. Let q, = 1 (k = 1, 2,, - 1), q = 3 i the Euclidea algorithm (1) for the relatively prime itegers a ad b. The ad a = F,- + F -, b = F + F, < W F ) a ( a, b M F + 2 J F + 1 ). If q. ^ 1 (k = 1, 2,, - 1), q > 3, the the correspodig itegers a ad b obey K the iequalities a > F ad b > F -. Proof. From the proof of Theorem 1, we have A - = F - ad B. = F whe * q k = 1 (1 ^ k < ). If q = 3, the by (4), A = 3 F 1 + F Q = F + 2F 1 = F _ L l + F ad, 9 similarly, J9 B = F + F. Sice F < F - < F, we ' have ad a = A < F _ L 1 + F = F _ L O b = B < F + F. = F ^ By (4) Similarly, Next, if q. ^ 1 (1 ^ k < ) ad q ^ 4, we have A > F 1 ad B 1 ^ F 9. K. x i. x u b = B > F + r a = A ^ 4A - + A ^ 4 F - + F = F + 2 F > F + F = F * +2 ' Lemma 4. Let the Euclidea algorithm for the itegers a ad b be (1), where q, > 2 f o r a t l e a s t t h r e e i d i c e s k err w h e r e q > 2, q > 3 f o r 1 < p m < p f m. The a > F ^ ^ o - Proof. Let q k > 2 for k = m, p (1 < m < p < ). The, parallelig the proof of Lemma 2, we obtai (5) a ^ A > F ^ + F F + F F +1 -m+1 m - 1 -p+1 p-1 " Now the last expressio is greater tha F + «provided ( 6 ) F,- F, + F. F > F. -m+1 m - 1 -p+1 p-1 Sice ± F -s + lvl > 2 F

6 1973] ON THE LENGTH OF THE EUCLIDEAN ALGORITHM 61 for 1 < s ^ by Lemma 1, the iequality (6) is valid. We coclude from (5) that a > A > F, - + F = F, If for some idex m, 1 ^ m ^, we have q ^ 3, the A. ^ F. for k = 1, 2, m K. K., m - 1 ad by (4) A > 3 F -, + F = F _ L i + F i > m m - 1 m-2 m+1 m - 1 F _ m+1 L l S A ^ (F + F ) + F > F + F = F m+1 v m+1 m - r m - 1 m+1 m m+2 e By iductioj A, > F. - for m ^ k <. Now so a > F J. +2 A > 2 A, + A > 2 F + F 1 = F _ L O The fial case to cosider is whe q = 2 for some idex m, 1 ^ m < ad q ^ M m M 3 8 A s i the proof of L e m m a 2? i t i s e a s i l y show that Thus, A, ^ F. + F - F. (k = m, m + 1, ' e, - 1). k k m - 1 k-m A > 3A - + A > 3F 1 + F + (3F - + F )F m - 1 -m-2 m - 1 > F + F + ( F + F )F > F l ~~ m+1 - m - 1 f r m * provided F j F -, + F - F > F -m+1 m m - 1 m This is the case sice, by Lemma 1, F F > I F -s+1 s~l 2 for 1 < s ^ ad, hece, F F + F F > (F + F ) > F r -m+1 m m - 1 m K - 2 ; -2 Therefore, a > F? i all cases cosidered i this Lemma. Collectig the results i the last three lemmas, we have proved the followig: Theorem 2. Let h> be the set of ordered pairs (a,b) such that (a,b)»(f $F - ). There are exactly + 1 pairs i E such that i(a,b) =. These pairs are obtaied from the formulas a = F +1 + F - m + l F m-1> b = F + F - m + l F m - 2

7 62 ON THE LENGTH OF THE EUCLIDEAN ALGORITHM Feb (m =, 1, 2,, ), where F = F 1 = ad F. for each j > is the j Fibo- -A - 1 " 31 acci umber (2). The results i Theorem 2 were suggested to the authors by cosiderig a umber of special cases o a IBM 36/65 computer. REFERENCES 1. R. L. Duca, "Note o the Euclidea Algorithm/' The Fiboacci Quarterly, Vol. 4 (1966), pp O. Perro, Die Lehre vo de Kettebruche, Vol. 1, Teuber, Stuttgart, J. V. Uspesky ad M. A. Heaslet, Elemetary Number Theory, McGraw-Hill, LETTERS TO THE EDITOR Dear Editor: I the paper (*) by W. A. Al-Salam ad A. Verma, "Fiboacci Numbers ad Euleria Polyomials," Fiboacci Quarterly, February 1971, pp , a e r r o r occurs i(9), which is readily corrected. I will geeralize their (4) by defiig a geeral polyomial operator M by (I) Mf(x) = Af(x + c t ) + Bf(x + c 2 ), c t c 2, where f(x) is a polyomial ad A, B, c l9 ad c 2 are give umbers. With D = d/dx, we ote that M = A e l D + Be 2 D so that or QO OO Mf(x) = A J^ T ^ W + B 2 = = T D * f(x) ' W (II) Af(x + C l) + Bf(x + c 2 ) = J^ - D f(x), = OO where W = Ac* 1 + BcL 2 is the solutio of W +2 i r = PW +1,- - ^ QW ad c \ 4 ^ c 2 are the roots T- L of x 2 = Px - Q. I (*), Eq. (4) is a special case of (I) with A = fi ad B = 1-1±, There are two cases of (II) to cosider: Case 1. A + B /, If A = B, we obtai from (II) V_ v (III) f(x + C l ) + f(x + c 2 ) = 2-5? D l l f ( x ) > = where V = 2, Vi = P, ad V, = PV,- - QV. If Cj ad c 2 are roots of x 2 = x + 1, u» i * [Cotiued o page 71. ]