A LIMITED ARITHMETIC ON SIMPLE CONTINUED FRACTIONS - II 1. INTRODUCTION

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1 A LIMITED ARITHMETIC ON SIMPLE CONTINUED FRACTIONS - II C. T. LONG J. H. JORDAN* Washigto State Uiversity, Pullma, Washigto 1. INTRODUCTION I the first paper [2 ] i this series, we developed certai properties of the simple cotiued fractio expasios of itegral multiples of quadratic surds with expasios of the form [a,b] or [a,b,c] where the otatio is that of Hardy Wright [1, Chapter 10]. For easy referece, we restate the priciple results here. Theorem 1. deote the k Let = [a,b], let be a positive iteger, let p, /q. coverget to let t, = q, - + q, - for k > 0 where we take q_ x = 0. The = [r/s] if oly if = <l 2i _ 2 > r = p 2m-2' s = t 0 2m-2 0 for some m > 1. Theorem 2. Let,, p, /q, t, be as i Theorem 1. The = [u,v,w] if oly if v = q 2 m _ l 9 vu = P 2 m _ 1 - If vw = t 2 m _ 1-2 for some iteger m ^ 1. * 8 th Theorem3. Let = [ a, b, c ], let p, /q, be the k coverget to, let t^ = q k _ 1 + q k + 1 s k = p k _^ + p k + 1 for k > 1. The, for every iteger r > 1, we have q 2 r * ^ = [ p 2r 9 fc 2r> c t 2 r / b ] ' q 2r-l = t p 2r-l " X ' l > Sr-i " 2 ] hv-l ' * = [ S 2r-1> q 2 r - l ' ( c ' + 4 c / b ) q 2 r - l J t 2 r ' ( = [ s 2 r - 1, 1, q 2 r - 2, 1, (be + 4)q 2 r - 2]. Of course, for a = b = c = 1, the precedig theorems give results ivolvig the golde ratio, (1 + V*5)/2, the Fiboacci Lucas umbers sice, i that case, *The first author was supported by NSF Grat GP

2 136 A LIMITED ARITHMETIC [March { = (1 + VB)/2. p k = F k + 2, q k = F k + 1, ^ = L k + 1, ^ = ^ where F L deote respectively the Fiboacci Lucas umbers. I the preset paper, we devote our attetio primarily to the study of the simple cotiued fractio expasios of positive ratioal multiples of quadratic surds with expasios of the form [a]. Agai, we ote that, for a = 1, the theorems specialize to results about the golde ratio Fiboacci Lucas umbers. 2. PRELIMINARY CONSIDERATIONS Let the itegral sequeces {f } {g } be defied as follows: ri>0 >0 (1) f 0 = 0, ft = 1, f = a ^! * ^ ' - (2) go = 2, gt = a, g = ag _ 1 + g ^, > 0, where a is ay positive iteger. These differece equatios are easily solved to give (3) f = * ", => 0, 7a (4) g = t +?, > 0, where = (a + Va 2 + T)/2 J = (a - */a 2 + 4)/2 are the two irratioal roots of the equatio (5) x 2 - ax - 1 = 0.

3 1970] ON SIMPLE CONTINUED FRACTIONS - H 137 Icidetally, if fl is a quadratic surd, we will always deote the cojugate surd by p. The followig formulas, of iterest i themselves, geeralize results for the Fiboacci Lucas umbers are easily proved by iductio, f = 2 > Z : ^(2! - M+ li) 2 a i + 1 i=0 ^ / (6) > 0, fo 2+l i=0 (7) % = f - l + W ^ > ( 8 ) f m + + l = f m f + W + 1> m ^ 0 ' * > ( 9 ) ^r-+l = f m^ + W + 1' m * ' ^ > (10) f f - f A ^ = (-l) m ~h JJt9 l < m <. m m-1 +1 m-+1' Also, we obtai i the usual way from (8) the followig lemma. Lemma 4. For the itegral sequece (f } ^ 0 we have that f If if oly if m, where m are positive itegers m > 2 if a = 1. 3, PRINCIPAL RESULTS Our first theorem, together with the results of the first paper i this couplet, yields a series of results cocerig the simple cotiued fractio expasio of multiples of = : [a] by the reciprocals of positive itegers* The theorem is also of some iterest i its ow right. Theorem 5. Let = (a + b*/c)/d with a, b, c, d itegers, c ot a perfect square, c d positive. Let r be a positive ratioal umber such that 2ar/d is a iteger. Let a 2 + d 2 = b 2 c let 1 < < r. The r = [a 0, a l8 a 2, ],

4 138 A LIMITED ARITHMETIC [March if oly if L 2 a r 1 r ' a " ~ c P a i > a 2' ' " 1 * Proof. We ote first that a^ - 2ar/d is p positive. This is so sice [ ar + r l W c ^ ra + rbvc" a 0 - ( 3 _^ "J! ^ c g. so that 2ar ^ - r a + rbn/c" d = 51 (*- - b 2 c > b's/c" U- rd 1 a + h\l~c r { - 1 => o, by hypothesis. Now let /x = [a^ a 2, a 3, ] so that The r = a t 2ar "1 1 0, a 0 -, a ls a 2, = 2 a r t 2ar r i + bvc 2a\ d ~ d J r(-a + b's/c) -d(a + b^f) r(a 2 - b 2 c) a + b^c" ~. dr = _ r

5 1970] ON SIMPLE CONTINUED FRACTIONS - H 139 the proof is complete. Corollary 6. Let a be positive itegers. Let = [a] let > f. The = [a 0, a l9 a 2, ] if oly if = [0, a 0, - a, a l9 a 2, " ' ]. Proof. Sice * * > a + \ / a f = [a] = j, we may use the precedig theorem with a = a, b = 1, c = a 2 + 4, d = 2, r =. The result the follows immediately sice is a iteger 2ar _ 2a d ~ 2 + d 2 = a = b 2 c, as required. Now for f = [a] = s - The covergets p, /q. are give by the equatios p 0 = a, P l = a 2 + 1, p = ap _ 1 + p _ 2, (11) ^ 2, q 0 = 1, q t = a, q = a q ^ + q 2, it is clear that p = f q = f - for ^ 0. Also, p' = f q f = f - for ^ 0, where p f /q T is the coverget to l/f. The

6 140 A LIMITED ARITHMETIC [March followig results could all be stated i terms of the sequeces (p } {q }; istead, we use the sequeces (f } ( g }. Corollary 7. Let r, s, be the positive itegers with > = r = f 2m' a d S = [ a ]. The f/ = [0, r, s], if oly if, = f 2 m - l ' q - for some m ^ 2. ^2m-l Proof. This is a immediate cosequece of Theorem 1 with a = b, Corollary 6. Corollary 8. Let u, v., w, be positive itegers with > = [ a ]. The f/ = [0, u, v, w ], if oly if, v = f 2m> vu = ^ m + l " 1 ' vw = g 2-2 for some iteger m ^ 2. Proof. This is a immediate cosequece of Corollary 6 Theorem 2 with u = v = w = a. The ext corollary results from Theorem 3 Corollary 6 by takig a = b = c. However, sice, i this special case, parts (a) (b) of Theorem 2 yield results already obtaied, we cocer ourselves oly with parts (c) (d). r > 1, Corollary 9. Let be a positive iteger greater tha. The for 4 = [ > < W f 2r' < a2+ ' 4 > f 2r] ^ = [0, g 2 r , f 2r+1-2, 1, (a* + 4)f 2r+1-2 ]. The ext theorem shows that the periodic part of the simple cotiued fractio expasio of for siy positive iteger > f = [a] is almost symmetric. Of course, by Corollary 6, the same thig is true of f/. Theorem 10. Let a be positive itegers with > = [a]. The = [a 0, a t,, a ] the vector (a l9 a 2,, a _-) is symmetric if r ^ 2.

7 1970] ON SIMPLE CONTINUED FRACTIONS - II 141 Proof. Sice a 0 = [ ], we have that 0 < f - fy < 1 f! = 1 - i, 1 ( - a 0 where f j is the first complete quotiet i the expasio of. Moreover, 1 1 fl = - a0 ^ + a 0 so that -1 < J" <0 f sice ao + /f is clearly greater tha oe, Thus, fj is a reduced quadratic surd by the geeral theory (see, for example [3, Chapter 4]) has a purely periodic simple cotiued fractio expasio, say fl = [ait a 2,, a r ]. Additioally, we also have that [a, a -, ', a-] is the expasio of the egative reciprocal of the cojugate of lo Thus, r. o i I, L a r*, a r_]_»., j a ', a- 1 J J = - j = 7- + a* v so that (12) J = [0, a r - a Q, a r - l f a r _ 2, -, a ^ a r ]

8 142 A LIMITED ARITHMETIC [March But, from above, = [a 0, &i,, a r ], by Corollary 6, (13) - = [0, a 0 - a, a lf a 2,, a r ]. Thus, comparig (12) (13), we have that the vector (a ls a 2,, a _-) is symmetric. We ow tur our attetio to the cosideratio of more geeral positive ratioal multiples of = [a]. Theorem 11. Let r be ratioal with 0 < r < 1. If the simple cotiued fractio expasio of r is ot purely periodic, the r = [0, a 1? a 2,, a j I _ r.., r ~ La " V V l ' a - 2 ' "» a 2 ' a J for some ^ 2. Proof. If r had a purely periodic simple cotiued fractio expasio, the r would have to be a reduced quadratic surd so that r > 1-1 ^ rt < 0. But the first of these iequalities implies that (14) j < r < 1,, sice = -l/f, the secod implies that (15) > r > 0

9 1970] ON SIMPLE CONTINUED FRACTIONS - H 143 which is already implied by (14). Therefore, sice r is ot purely periodic, we have (16) 0 < r < i, so that 0 = [ r ] = a 0. Now cosider f 1 = ^ > 1. set aj = [ i] ^ 1. Agai, 1 1 _ f i - aj = h > 1 (2 = " T T~~ - H 7 + at sice?" = - 1. Therefore, - 1 < J 2 < 0 2 i s a purely periodic simple cotiued fractio expasio, reduced. Thus, 2 a s 6 = [ a 2i a 3,, a ], r = [0, a l5 a 2,, a j, as claimed. Also, + ai = - - = a, a -,, a 0, r * T- L ' -1 ' 2 J ' * 2

10 144 A LIMITED ARITHMETIC [March so that the proof is complete. "~" ~~ I a a., a.., a ~ «a^, a i r L 1-1 r-2 2 J Theorem 12. Let r be ratioal with 0 < r < 1. If the simple cotiued fractio expasio of rf is purely periodic, the r = [ v a i» "» *J for some > 0. - = fa, a -,, a A 1 r L -1 0 J Proof. Sice the simple cotiued fractio expasio of rf is purely periodic, it is reduced we have by the precedig proof that < r < 1 Sice we also have f i r - - i.2. = = a, a.,, a~ J, r T L -1 the proof is complete. I passig, we ote that the periodic part of the expasios of rf eed ot exhibit ay symmetry or eve ear symmetry. For example, for a = [1] = 2, we have that

11 1970] ON SIMPLE CONTINUED FRACTIONS - II 145 o r = [0, 2, l f 3, 1, 1, 3, 9] a = [1, 4, 1, 2, 6, 2]. Also, it is easy to fid ratioal umbers r with 0 < r < 1 such that the surds va a/v are ot equivalet where we recall that two real umbers JJL v are said to be equivalet if oly if there exist itegers a, b, c, d with ad - be j = 1 such that u, = SL4. ^ cv + d However, as the followig theorems show, there exist iterestig examples, where ear symmetry of the periodic part of the expasios of r f equivalece of r r/f both hold. We will idicate that r f f/r are equivalet by the otatio / r b r Theorem 13. Let a be a positive iteger, let f = [a], let the sequeces (f } {g } be as defied above. The, for > 1, ^O -0 >0 f 2+l f 2+2 = [a Q, a x,, a r ] 2+2 r.. o 1 7 = a, a,? ", a J r f 2-KL ^ where the vector (a 2, a 3, *, a r, ao) is symmetric, a 0 = a 2 = 1, aj = f - 1 l o *4+3

12 146 A LIMITED ARITHMETIC [March Proof. We first demostrate the purely periodic ature of the expasios i questio. From the defiitio, it i s clear that f i s strictly icreasig for ^ 2. Also, f /f - is the coverget to l/f. Therefore, (17) ^ L. < i <^ ±1< 1. i 2+l * x 2+2 it follows from the proof of Theorem 11 that f f 2 - /fl 2 has a purely periodic expasio. Also, from Theorem 12, f 2 2 /f 2 - has a purely periodic expasio whose period is the reverse of that for f 2 +l ^ 2 +2* Additioally, from (17), it follows that < f 2+l _ i < 1 / f 2+l _ f 2 \ = _ 1_ f 2+2 f 2 \ f 2+2 f 2+l/ 2f 2+l f! 2+l 2+2 so that -. f 2+l > _ g,, 1 < ^ _. < _ + 1 *2+2 x 2+l 2+2 < 2 f t a y + 1 < 2 2+l 2+2 Thus, a 0 = [ff 2+1 / f ] = 1. Now 2+l. f.! f 2+2 we claim that W 4+3 * - K ^ ^1! < f so that a 4 = f 4-1. To see that this is so, we ote mat, sice f 2 /f - is the coverget to,

13 1970] ON SIMPLE CONTINUED FRACTIONS 147 f 2+2 ^ f4+4. < f f " < ^ 2+l ^ f2+l f 4+4 ^ f2+l 1 < ^ <. % But this gives, usig (10), 2-fl 2+l * f x > f f f f 2+2 f 4+3 f 4+3 o r (18) < f 4+3 a s desired. Also, we have that f 2+2 t f 4+3 f * "^ f 2+l 4+2 so that, agai by (10), o < ^ S J i. f i *2+l f 4+3 ~ f 2+2 f 4+2 ^, :, f 2+l f 2+2 f f 2+l f 2+l f 4+3 f 2+l

14 148 A LIMITED ARITHMETIC [March Thus, aj = [f i] = f. - 1 as claimed. Fially, to show the symmetry of the vector (a 2, &$,, a, a Q ), it suffices to show that (19) 1 - =. «. 2+l a 2 f 2+l f7~7 2+2 " * " a 0 a t Makig use of the determied values of a 0 aj settig a 2 = 1, meas that we must show that this (20) 1 = **** f 2+l f 2+l f ( ' 1 x 2+2 " < f «which will also, of course, cofirm the fact that a 2 = 1. Now (20) is true if oly if

15 1970] ON SIMPLE CONTINUED FRACTIONS - H _ f 2+l 2+2 ( f " «x 2+2 which i s true if oly if 1 f 2+l. x (f 4+3 " 1) - ^ ' f + f f x 2+l * 2+2 which, i tur, is true if oly if f H s f - ^ f 2 + l - f " f 2 + l + to+3 ^ " " To see that this l a s t equatio i s true we make use of (8), (10), the fact that 2 = ^ + i 0 obtai f 2+2 ^f2+2 = f 2+2 f 2+l + * f 2+2 " ^ f2+l f ^f2+2 f 2+l " f " f 2-KL + ^ 2 Q + 2 " f V < - l f W " ^ ^ " a^f2+l f 2+2 f f 2+l + a f f 2+l f 2+2 " f f 2+2 f (f + f ) 2+2 u ; f f - f l f f f + f f - f 2-1 x 2+2 I l f - 1 I 4+3

16 150 A LIMITED ARITHMETIC [March This completes the proof. Because of the similarity of method, the followig theorems a r e stated without proof. The otatio is as before. T h e o r e m 14. F o r > 2, 2 *. r i 7 = L a 0> a i> a 2» "» a r J 2+l f 2+l ^ r -i i -f - = La 3-1, a 4, a 5,, a r, a 2, a 3 J, 2 where the vector (a 3, a 4,, a r ) is s y m m e t r i c, a 0 = 0, a 4 = 1, a 2 f 4+l " l f a d a 3 = f 3 +! T h e o r e m 15. Let >: 2 be a iteger. The +2 r.., f = [a 0, a t,, a r J, 11 > _ r»... " 1?""" * ' ~" L a r' +2 a r - l ' ' a o J s - ~ e f = [ b 0, b j,, b s ], T^-t = [ V V l ' ' fc o] T h e o r e m 16. Let be a positive iteger. The g 2+l t T.., f = [ a 0, a A, a 2,, a r J, 8 2+2

17 1970] ON SIMPLE CONTINUED FRACTIONS - II 151 g l = [ 2 9 a 4? a 5,.., a. a 2, a 3 ] r the vector (a 3, a 4,», a r ) is symmetric with a 0 = 0, a, t = 1, f 4+3 " l s a d a 3 = 3e Theorem 17, Let be a positive iteger. The g 2 =2+l = [a 0, *U, a r ] g 2+i g 2 = [ a r, a r _ l S, a if a 0 ], where the vector (a 29 a 3, «, a r s a 0 ) is symmetric with a^ - a 2 = l f a l = f 4+l " X «I view of the precedig results 9 oe would expect a iterestig theorem cocerig the simple cotiued fractio expasio of i.. s but we were ot able to make a geeral assertio value for all a. To illustrate the difficulty, ote that s whe a = 2 = 1 + *J"2 9 we have f 4 = [ 0, 1, 5, 1, 3, 5, 1, 7], &4 f g5 f = [0, 1, 5, 1, 5, 3, 1, 4, 1, 7],

18 152 A LIMITED ARITHMETIC [March f 1- = [0, 1, 5, 1, 4, 1, 3, 5, 1, 4, 1, 7]. &6 However, for i; = = [1] = «^ -» we obtai the followig rather elegat result: Theorem 18. Let a = (1 + *v/5)/2 let F L deote the Fiboacci Lucas umbers, respectively. The, for > 4, F (21) _. a = [0, 1, 2, 1,, 1, 3, 1,, 1, 4] L L (22) -I- a = [ 3, 1,, 1, 3, 1, -, 1, 2, 4 ], where, i (21), there are - 4 oes i the first group - 3 oes i the secod group just the reverse i (22). Proof. Set x = [2, 1, -., 1, 3, 1,, 1, 4] = [2, 1,, 1, 3, 1,, 1, 4, x ]. The it is easy to see by direct computatio as o computes covergets, that a x + b \ JT7T where

19 1970] ON SIMPLE CONTINUED FRACTIONS - H 153 a " 4 ( L - l F - l + F -2> 2 + ( L -2 F -l + F -3 F -2> = 4 F + F F - l + <-»*' b = L - l F - l + F -2 = F C " 4 ( L - l F F -2 F -4> + < L -2 F -3 + V s W = 4F -l + F F - 3 Moreover, from (23), d = L F + F F = F 2 V l x (a - d ) + J ( a - d ) 2 + 4b c 7 y 2c y = [o, i. x j X x + 1 (a - d ) + J ( a - d ) 2 + 4b c y (a - d +2c )+ (a - d ) g +4b c ' = (a - d -2b ) + J(a - d ) 2 + 4b c J 2(a - b + c - d ) Now

20 154 A LIMITED ARITHMETIC [March a - d - 2b = 4F 2 + F F, + ( - l ) - F 2-2F = 2 F 2 + F F, - F F (25) = F (2F + F - F Q) ' F ^F+2 " F - 2 * = F L, a - b + c - d v = (a - d - 2b ) + b + c ' (26) = F L + F 2 + 4F 2 + F F F L + 2F -L - 1 = L 2 < a " V V = < 4 F + F F - l + ^ " p Ll>' + 4 F ^ ( 4 F ^ _ 1 + F F _ 3 ) (27) = K F l F ^ 4 F L l + F F a-3> = F < F F - l + 4 F F - 3 > = 5 F 2 L 2 T h u s, usig (25), (26), (27), i (24), we obtai Y F L + F L \/5 F - J / = = ji 1 + V5 = 2 " 2L2 L '

21 1970] ON SIMPLE CONTINUED FRACTIONS H 155 as claimed. The other part of the proof is a immediate cosequece of Theorem 11. Fially, we commet o the questio of the equivalece of rf f/r. If r = g ^ /f or r = g m /g^9 where m are oegative itegers, it frequetly turs out to be the case that rf ~ f/r. However, this is ot ecessarily the case hece, & fortiori, it is.ot ecessarily the case for more geeral r. For example, for a = (l+\[5)f2 = [1], J. a = [0, 1, 2, 3, 1, 4] 1» a = [ 3, 1, 3, 2, 4] where 3 = f 4 = g 2 7 = g 4 ; other examples are easily foud. However, if r = f s = f for oegative itegers m the we always have s * r * as the followig theorem shows. Theorem 18. If m are oegative itegers, the f f m *. t m

22 156 A LIMITED ARITHMETIC [March Proof. Without loss of geerality, we may assume that 0 < m < that (m,) = 1. We let f f f f = m 2 3 m + 2 b = c = f d = -HJ D c 2qm+l' a J ' ^ q m + l ' f ^ m where q is chose so that 2q + 2 = 0 (mod ), as may easily be doe sice (m,) = 1. With this choice for q it follows f from Lemma 4 that l f 2 G im+2 ^ f ml f 2am s o t a t a» k» c» a d d a r e a^itegers. Also, by (10), ad - be = J*L*E1±2. V p q m _ f2 f f 2qm+l m ^ = f f - f 2 2qm+2 2qm 2qm+l = -1. Fially, we show that f (28) ^ f / f a F " - \ m \ m + b H + c for this choice of a, b, c, d. Makig the idicated substitutios, we have that (28) holds if oly if f m f 2qm+2 / f J\ + f f I F " ' * / f 2qm+l T~ m " * t \ m / = 7 T \ TT 0 '* / f \ ' f: f / t. I j. 2 q m

23 1970] ON SIMPLE CONTINUED FRACTIONS - II 157 this is true if oly if ^ f2qm+l + ^f2qm ^f2qm+2 + f 2qm+l e But this is clearly true sice af 2 = a + 1 af 0, - + f = f 0, 0 J * 2qm+l 2qm 2qm+2 the proof is complete, Fially 8 we ote that the list of stated theorems is ot exhaustive. could o doubt prove theorems cocerig Oe f f f, y y? " s 9 f» f so o. However? we were ot able to arrive at geeral formulatios of the expasios of f f g m t m t r ' f ' T" 8^f o r T" * & ^ m & t 9 for arbitrary positive itegers m. The results stated seem to be the most iterestig. REFERENCES 1. G. H. Hardy E. M. Wright, A Itroductio to the Theory of Numbers, Oxford Uiversity Press, Lodo, C. T. Log J, H, Jorda 9 "A Limited Arithmetic o Simple Cotiued Fractios," Fiboacci Quarterly, 5 (1967), pp C. D. Olds, Cotiued Fractios, Rom House, New York, 1963.