Minimal multipliers for consecutive Fibonacci numbers

Transcription

1 ACTA ARITHMETICA LXXV3 (1996) Minimal multipliers for consecutive Fibonacci numbers by K R Matthews (Brisbane) 1 Introduction The Fibonacci and Lucas numbers F n, L n (see [2]) are defined by Since F 1 = F 2 = 1, F n+2 = F n+1 + F n, n 1; L 1 = 1, L 2 = 3, L n+2 = L n+1 + L n, n 1 (1) F n 1 F n F n 2 F n+1 = ( 1) n, it follows that gcd (F n, F n+1,, F n+m 1 ) = 1 for all m 2 Consequently, integers x 1,, x m exist satisfying x 1 F n + x 2 F n x m F n+m 1 = 1 We call (x 1,, x m ) a multiplier vector By equation (1), one such vector is (2) M n = (( 1) n F n 1, ( 1) n+1 F n 2, 0,, 0) The problem of finding all multiplier vectors reduces to finding a Z-basis for the lattice Λ of integer vectors (x 1,, x m ) satisfying x 1 F n + x 2 F n x m F n+m 1 = 0 It is easy to prove by induction on m that such a lattice basis is given by M n+2, L 1,, L, where (3) L 1 = (1, 1, 1, 0,, 0), L 2 = (0, 1, 1, 1, 0,, 0), L = (0,, 0, 1, 1, 1) Hence the general multiplier vector has the form M n + y 1 L y L + y m 1 M n+2, where y 1,, y m 1 are integers [205]

2 206 K R Matthews Table 1 Least multipliers, m = 4, 5, 2 n 20 n Least multipliers, m = n Least multipliers, m = A recent paper by the author and collaborators [1] contains an algorithm for finding small multipliers based on the LLL lattice basis reduction algorithm Starting with a short multiplier, we then use the Fincke Pohst algorithm to determine the shortest multipliers When applied to the Fibonacci sequence, this experimentally always locates a unique multiplier of least length if n > 1 For m = 2, it is well known that the extended Euclid s algorithm, applied to coprime positive integers a, b, where b does not divide a, produces a multiplier vector (x 1, x 2 ) satisfying x 1 b/2, x 2 a/2, which is consequently the unique least multiplier With a = F n+1, b = F n, n 3, this gives the multiplier vector M n However, for m 3, the smallest multiplier problem for F n,, F n+m 1 seems to have escaped attention (Table 1 gives the least multipliers for m = 4 and 5, 2 n 20) In this paper, we prove that there is a unique multiplier vector of least length if n 2, namely W n,m, where (4) W n,m = ( 1) n V n,m = ( 1) n (W n,1,m, W n,2,m,, W n,m,m ), which is defined as follows, using the greatest integer function: Let (5) (6) P n = (F n 1, F n 2, 0,, 0), V n,m = P n G n,1,m L 1 + G n,2,m L 2 + ( 1) m G n,,m L, where the nonnegative integers G n,1,m,, G n,,m are defined as follows:

3 Let Minimal multipliers 207 Fm r (F n 2 + F r ) H n,r,m =, 1 r m Then for m even, Hn,r,m if 2 r m 2, r even, (7) G n,r,m = if 1 r m 3, r odd, H n 1,r+1,m while for m odd, Hn,r,m 1 = G (8) G n,r,m = n,r,m 1 if 2 r m 3, r even, H n 1,r+1,m+1 = G n,r,m+1 if 1 r m 2, r odd The definition of G n,r,m extends naturally to r = 1, 0, m 1, m: G n, 1,m = F n 3, G n,0,m = F n 2, G n,m 1,m = G n,m,m = 0 Then equations (4) (6) give (9) W n,r,m = G n,r 2,m + G n,r 1,m G n,r,m It was not difficult to identify these multipliers for 2 n 2m + 2 (see Table 2) It was also not difficult to identify them for m even, n arbitrary, though the initial form of the answer was not elegant However, it did take some effort to identify the case of m odd, n arbitrary This was done with the help of the GNUBC 103 programming language, which enables one to write simple exact arithmetic number theory programs quickly To prove minimality of length, we use the slightly modified lattice basis for Λ, L 1,, L, W n+2,m, which is the one always produced by our LLL-based extended gcd algorithm We then have to prove that if n > 1, x 1 L x L + x m 1 W n+2,m W n,m 2 W n,m 2 for all integers x 1,, x m 1, with equality only if x 1 = = x m 1 = 0 The proof divides naturally into two cases If x m 1 is nonzero, the coefficient of x 2 m 1 dominates For this we need Lemmas 5 and 11 If x m 1 = 0, the argument is more delicate and divides into several subcases, again using Lemma 5 The other lemmas play a supporting role for the derivation of Lemmas 5 and 11 In particular, Lemma 2 is important, as congruence properties in Lemma 4 reduce the calculation of the discrepancies for general n to the case n 2m + 2, where everything is quite explicit 2 Explicit expressions for W n,r,m For later use in the proof of Lemma 11, we need the following simpler form for W n,1,m in terms of the least integer function:

4 208 K R Matthews Lemma 1 Fn+m 3 F F W n,1,m = m Fn if m is even, if m is odd P r o o f The identity F n+m 3 = F n 1 F F n 3 follows from the well known identity F a+b = F a F b+2 F a 2 F b, with a = m and b = n 3 Consequently, if m is even, F (F n 3 + 1) W n,1,m = F n 1 G n,1,m = F n 1 Fn+m 3 + F Fn+m 3 F = =, and similarly if m is odd The W n,r,m with m even and 3 n 2m + 2 have an especially simple description in terms of Lucas numbers and play a central role in the proof of Lemma 5: Lemma 2 (See Table 2, which summarizes (a) and (c)) (a) Let 3 n m + 2, m even If n is odd, Ln r 2 if r n 3, W n,r,m = 1 if r = n 2, n 1, 0 if r n If n is even, L n r 2 if r n 4, 2 if r = n 3, W n,r,m = 1 if r = n 2, 0 if r n 1 (b) Let 3 n m + 2, m odd Then W n,r,m = W n,r,m 1, 1 r m 1, W n,m,m = 0 (c) Let 3 + m n 2m + 2 Then Ln r 2 if r 2m n + 3, W n,r,m = L n r if r = 2m n + 3 These formulae follow from explicit expressions below for G n,r,m in terms of the Fibonacci numbers, when m is even: Lemma 3 (a) Let 3 n m + 2 If r is even, Fn r 2 if r n 2, G n,r,m = 0 if r n 1

5 Minimal multipliers 209 Table 2 The W n,r,m, 1 n 2m + 2, m even n W n,1,m W n,2,m W n,3,m W n,4,m W n,5,m L L L L 2 L 1 L L 3 L 2 L L L 4 L 3 L 2 L 1 L 0 1 m 1 L m 4 L m 5 m L m 3 L m 4 m + 1 L L m 3 m + 2 L m 1 L m + 3 L m L m 1 m + 4 L m+1 L m m + 5 L m+2 L m+1 m + 6 L m+3 L m+2 2m L 2m 3 L 2m 4 L 2m m + 1 L 2 L 2m L 2m 4 2m + 2 L 2m L 2 L 2m 3 Table 2 (cont) n W n,6,m W n,m 3,m W n,,m W n,m 1,m W n,m,m m 1 L m L L m + 1 L 2 L 1 L m + 2 L 3 L 2 L L 0 1 m + 3 L 4 L 3 L 2 L m + 4 L 5 L 4 L L 2 m + 5 L 6 L L 4 L 3 m + 6 L L 6 L 5 L 4 2m L m+1 L m L m 1 L 2m + 1 L m+2 L m+1 L m L m 1 2m + 2 L m+3 L m+2 L m+1 L m

6 210 K R Matthews If r is odd, G n,r,m = Fn r 4 if r n 3, 0 if r n 2 (b) Let m + 3 n 2m + 2 If r is even, Fn r 2 if r 2m n + 2, G n,r,m = F n r 2 F n 2m+r 2 if r 2m n + 3 If r is odd, G n,r,m = Fn r 4 if r 2m n + 1, F n r 4 F n 2m+r 2 if r 2m n + 2 (c) If n = 1 or 2, G n,r,m = 0 for 1 r m P r o o f We assume r and m are even, as the case of odd r depends trivially on this case We start from the following identity, valid for a even: (10) F a r F b F a F b r = ( 1) r F b a F r Then Hence (11) and (12) (13) r F n 2 F n r 2 = F n F r r (F n 2 + F r ) = F n r 2 + ( r F n )F r = F n r 2 + ( r + ( 1) n n+2 )F r Fm r (F n 2 + F r ) (Fm r F n )F r G n,r,m = = F n r 2 + (Fm r + ( 1) n n+2 )F r = F n r 2 + (a) Assume 3 n m + 2 First suppose r n 2 Then m r m n and hence and Hence 0 n+2 r 0 r + ( 1) n n+2 2 r 0 ( r + ( 1) n n+2 )F r 2 rf r < 1, as 2 r F r < if 2 r m 2 Hence (Fm r + ( 1) n n+2 )F r = 0 and equation (13) gives G n,r,m = F n r 2

7 Next suppose r n 1 Then Minimal multipliers r(f n 2 + F r ) r(f r 1 + F r ) = rf r+1 < 1, where we have used the inequality F a F b < F a+b 1 if 2 a, 1 b Hence G n,r,m = 0 (b) Assume m + 3 n 2m + 2 First assume r 2m n + 2 Then m r n m 2 0 and r F n 0 and as seen before, the second integer part is zero in formula (12) and G n,r,m = F n r 2 Next assume r 2m n+3 Again we use a special case of equation (10): F n F r F n 2m+r 2 = ( 1) n r F 2m n+2 This, together with equation (12), gives Fm r (F r + ( 1) n+1 F 2m n+2 ) G n,r,m = F n r 2 F n 2m+r 2 + But r 2m n + 3 implies F r F 2m n+2 0 and as before, the integer part vanishes and we have G n,r,m = F n r 2 F n 2m+r 2 3 The discrepancies D n,r,m and E n,r,m Lemma 4 Let D n,r,m = W n,r 2,m W n,r 1,m W n,r,m, 3 r m (a) If n = t + 2Nm, m even, F n = F t + F Nm L Nm+t and D n,r,m = D t,r,m (b) If m is odd, then Dn,r,m 1 if r is even, D n,r,m = if r is odd D n,r,m+1 P r o o f (a) Let n = t + 2Nm, m even Then F n F t = F t+2nm F t = F Nm+t+Nm F Nm+t Nm = F Nm L Nm+t, as Nm is even Noting that F Nm 0 (mod ), we have from equation (7) and the definition of H n,r,m : G t,r,m + rf Nm L Nm+t if 2 r m 2, r even, F (14) G n,r,m = m G t,r,n + r 1F Nm L Nm+t if 1 r m 3, r odd

8 212 K R Matthews Hence, noting that Z = (F Nm L Nm+t )/ is an integer, we verify that with r even, equations (7) and (9) give Hence W n,r,m = W t,r,m + r+2 Z, W n,r 1,m = W t,r,m r Z + 2 r+2 Z, W n,r 2,m = W t,r,m + r+4 Z D n,r,m D t,r,m = ( r+4 ( r + 2 r+2 ) r+2 )Z = ( r + r+4 3 r+2 )Z = ( r + r+3 2 r+2 )Z = ( r + r+1 r+2 )Z = 0 Similarly for r odd (b) Let m be odd Then if r is even, equations (8) and (9) give and consequently W n,r,m = G n,r 2,m 1 + G n,r 1,m+1 G n,r,m 1, W n,r 1,m = G n,r 3,m+1 + G n,r 2,m 1 G n,r 1,m+1, W n,r 2,m = G n,r 4,m 1 + G n,r 3,m+1 G n,r 2,m 1, D n,r,m = 3G n,r 2,m 1 G n,r,m 1 G n,r 4,m 1 = D n,r,m 1 Similarly when r is odd, we find D n,r,m = D n,r,m+1 Lemma 5 For each n, we have W n,r 2,m = W n,r 1,m + W n,r,m, with at most three exceptional r, which satisfy D n,r,m = 1 P r o o f If m is even, Lemma 4(a) reduces the problem to the range 1 n 2m, where it is evidently true, by virtue of the explicit formulae for W n,r,m given in Lemma 2 We also observe that if m is even, then for n even, there are at most 2 odd r and one even r for which D n,r,m = 1 Then Lemma 4(b) gives the result when n is even Similarly for n odd As a corollary, we have Lemma 6 For fixed n > 1 and m 3, W n,r 1,m W n,r,m if 2 r m P r o o f This is clear when n m + 2, while for n m + 3, it is a consequence of the inequalities W n,m,m 1, W n,m 1,m W n,m,m and W n,r 2,m W n,r 1,m + W n,r,m 1, 3 r m We will need an alternative Z-basis for the lattice Λ Lemma 7 The vectors L 1,, L, W n+2,m form a Z-basis for Λ

9 Minimal multipliers 213 P r o o f This is a consequence of the identity W n+2,m = M n+2 + ( 1) n ( 1) i G n+2,i,m L i, which follows from equations (4) (6) Lemma 8 Let E n,r,m = G n+2,r,m G n+1,r,m G n,r,m, where m is even (a) If n t (mod 2m), then E n,r,m = E t,r,m (b) If n = 1, 2, m + 1 or m + 2, then E n,r,m = 0 for 1 r m If 3 n m and n is even, 1 if r = n 3, E n,r,m = 0 otherwise If 3 n m and n is odd, If m + 3 n 2m + 2, E n,r,m = E n,r,m = 1 if r = n 1, 0 otherwise 1 if r = n 2 m, 0 otherwise P r o o f (a) follows from equation (14), while (b) follows from the explicit form of G n,r,m given in Lemma 3 Lemma 9 Let E n,m = W n+2,m + W n+1,m W n,m Then (15) E n,m = (0,, 0) or ± L i for some i P r o o f By equations (4) (6), E n,m = ( 1) n (V n+2,m V n+1,m V n,m ) = ( 1) n ( 1) r E n,r,m L r r=1 If m is even, Lemma 8 gives the result directly, while if m is odd, we see from equation (8) that En,r,m 1 if 2 r m 3, r even, E n,r,m = if 1 r m 2, r is odd E n 1,r+1,m+1 Then Lemma 8, with m replaced by m 1 and m + 1, respectively, gives the result As a corollary, we have Lemma 10 For fixed r and m, W n+1,r,m W n,r,m

10 214 K R Matthews P r o o f If n m, the result follows from Lemma 2 If n m + 1, m even, or n m + 2, m odd, we have W n,r,m 1 and equation (15) gives W n+2,r,m W n+1,r,m + W n,r,m 1, which gives the desired result The remaining cases are simple exercises 4 A size estimate for W n+2,m Lemma 11 If n 5, (16) W n+2,m 2 > 2W n+2,m W n,m + 18 P r o o f W n+2,m W n,m W n+1,m = 0 or τ i L i, where τ i = ±1 Hence W n+2,m 2 2W n+2,m W n,m + W n,m 2 W n+1,m 2 2W n+1,m L i + L i 2 W n+1,m 2 3 Hence the desired inequality will follow if we can prove (17) W n+1 2 > W n But W n+1,r,m W n,r,m 0 and from Lemma 1, it is easy to prove that W n+1,1,m > W n,1,m + 3 if n 7 Then because W n,1,m 2 if n 4, inequality (17) follows if n 7 Cases n = 5 and 6 of inequality (16) can be verified using the following identities: W 5,m = ( 3, 1, 1, 1, 0, 0, ), W 6,m = (4, 3, 2, 1, 0, 0, ), W 7,m = ( 7, 4, 3, 1, 1, 1, ), W 8,m = (11, 7, 4, 3, 2, 1, ), if m 6 Also W 5,5 = ( 3, 1, 1, 1, 0), W 6,5 = (4, 3, 2, 1, 0), W 7,5 = ( 7, 4, 3, 2, 0), W 8,5 = (11, 7, 4, 4, 1), W 5,4 = ( 3, 1, 1, 1), W 7,4 = ( 7, 4, 3, 2), W 6,4 = (4, 3, 2, 1), W 8,4 = (11, 7, 5, 3), W 5,3 = ( 3, 2, 0), W 6,3 = (4, 4, 1), W 7,3 = ( 7, 6, 1), W 8,3 = (11, 10, 2) 5 The proof of minimality Theorem For all integers x 1,, x m 1, (18) x 1 L x L + x m 1 W n+2,m W n,m 2 W n,m 2, with equality only if x 1 = = x m 1 = 0

11 P r o o f Inequality (18) is equivalent to Minimal multipliers 215 (19) x 1 L x L + W n+2,m x m L i W n,m x i 2W n+2 W n,m x m 1 0 The left hand side of this inequality expands to L i L j x i x j + W n+2,m 2 x 2 m L i W n+2,m x i x m 1 j=1 Now ε i = L i W n,m = ( 1) n D n,i+2,m But by Lemma 5, D n,r,m = 0 for 3 r m, with at most three exceptional r, in which case D n,r,m = ±1 Also η i = L i W n+2,m = 0, with at most three exceptions Also 3 if i = j, 0 if j = i + 1, L i L j = 1 if j = i + 2, 0 if j i + 3 Substituting all this in the expanded form of inequality (19) gives the equivalent inequality (20) Q = Q 1 + Ax 2 m 1 2Bx m η i x i x m 1 2 ε i x i 0, where A = W n+2,m 2, B = W n+2,m W n,m and m 4 Q 1 = 3 x 2 i 2 x i x i+2 Now let x 1,, x be integers, not all zero We prove Q > 0 C a s e 1: x m 1 nonzero The coefficient of x 2 m 1 dominates For completing the square gives the equivalent inequality (21) ( Q = Q 2 + A + η 2 i (x i ε i + η i x m 1 ) 2 ) ( x 2 m 1 2 B ε i η i )x m 1 ε 2 i 0, where Q 2 0 Here 2x 2 1 if m = 3, 2x Q 2 = x 2 2 if m = 4, (x 1 x 3 ) 2 + x x x 2 3 if m = 5, T + x x x 2 m 3 + x 2 if m 6,

12 216 K R Matthews where m 4 (22) T = (x i x i+2 ) 2 But ε 2 i 3, η 2 i 3, ε i η i 6 Hence inequality (21) holds with strict inequality, if we can prove (A 3)x 2 m 1 3 > 2(B + 6) x m 1 This will be true if A > 2B + 18 and this follows from Lemma 11 if n 5 Finally, only the case n = 4 needs any thought and this is straightforward, as W 2,m = (2, 1, 0, 0,, 0), W 4,m = (4, 3, 2, 1,, 0), if m 4 C a s e 2: x m 1 = 0 The argument is more delicate We start by assuming m 6 Then Q = T + S 0 U, where and S 0 = 2x x x x 2 m 4 + 2x 2 m 3 + 2x 2 U = 2 ε i x i In what follows, we make use of the inequalities x(x ± 1) 0, x(x ± 2) 1 if x Z Clearly we need only consider T 3 C a s e 2(a): T = 0 Then Q = S 0 U and x 1 = x 3 =, x 2 = x 4 = Hence one of x 1, x 2 must be nonzero and one of x m 3, x must be nonzero Then a consideration of the possible terms in U shows that Q = S 0 U = 2x x x x 2 m 4 + 2x 2 m 3 + 2x 2 U 1 C a s e 2(b): T = 1 Then Q = 1 + S 0 U and there exists i such that x i x i+2 = 1, while x j = x j+2 if j i A consideration of the possible terms in U shows that S 0 U 0 and hence Q = 1 + S 0 U 1 C a s e 2(c): T = 2 Then Q = 2 + S 0 U If one of x 1, x 2, x m 3, x is nonzero, then S 0 2 and Q 1 Suppose x 1 = x 2 = x m 3 = x = 0 Then S 0 = x x 2 m 4 If at least two of the variables are nonzero, then S 0 2 and Q = 2 + S 0 U 1 If precisely one variable x k has nonzero coefficient, then

13 Minimal multipliers 217 S 0 = x 2 k If x k 2, we are done However, if x k = 1, then nonzero terms in U can contribute at most x 2 k ± 2x k 1 to S 0 U and we have Q = 2 + S 0 U 1 C a s e 2(d) T = 3 Then Q = 3 + S 0 U 0 Moreover, Q = 0 implies S 0 U = 3 and there exist indices I, J, K satisfying 3 I < J < K m 4 with x I = ε I = ±1, x J = ε J = ±1, x K = ε K = ±1, while x i = 0 if i I, J, K A consideration of cases shows this would in turn imply T 4 Finally, there remain the cases m = 3, 4, 5 m = 3: Here x 2 1 > 0 and Q = 3x 2 1 2ε 1 x 1 = x 2 1 2x 1 (x 1 ε 1 ) > 0 m = 4: Here x x 2 2 > 0 and Q = 3x x 2 2 2ε 1 x 1 2ε 2 x 2 = x x x 1 (x 1 ε 1 ) + 2x 2 (x 2 ε 2 ) > 0 m = 5: Here x x x 2 3 > 0 and Q = (x 1 x 3 ) 2 + 2x x x 2 3 2ε 1 x 1 2ε 2 x 2 2ε 3 x 3 = (x 1 x 3 ) 2 + x x 1 (x 1 ε 1 ) + 2x 2 (x 2 ε 2 ) + 2x 3 (x 3 ε 3 ) 0 Moreover, equality implies x 1 = x 3 = ε 1 = ε 3 and x 2 = 0 Then 0 = ε 1 ε 3 = (L 1 L 3 ) W n,5 = (1, 1, 2, 1, 1) W n,5 = ( 1) n (W n,1,5 W n,2,5 + 2W n,3,5 + W n,4,5 W n,1,5 ) But Lemma 6 implies W n,r 1,5 W n,r,5 if n > 1, r 2 Also Lemma 10 gives W n,3,5 W 5,3,5 = 1 if n 5 Hence we get a contradiction if n 5 Also we cannot have n < 5, as (ε 1, ε 3 ) = (1, 0), (0, 0), (1, 0) for n = 2, 3, 4, respectively Acknowledgements In conclusion, the author wishes to thank Dr George Havas for suggesting that the LLL-based extended gcd algorithm be applied to the Fibonacci numbers Calculations were performed using the GNUBC 103 exact arithmetic calculator program, together with a number theory calculator program called CALC, written by the author The author is grateful to the referee for painstaking efforts and for suggesting that the paper be revised to improve its readability

14 218 K R Matthews References [1] G Havas, B S Majewski and K R Matthews, Extended gcd algorithms, to appear [2] V E Hoggatt Jr, Fibonacci and Lucas Numbers, Houghton Mifflin, Boston, 1969 Department of Mathematics University of Queensland Brisbane, Australia Q krm@axiommathsuqozau Received on and in revised form on (2752)