ON SUMS AND RECIPROCAL SUM OF GENERALIZED FIBONACCI NUMBERS BISHNU PADA MANDAL. Master of Science in Mathematics

Transcription

1 ON SUMS AND RECIPROCAL SUM OF GENERALIZED FIBONACCI NUMBERS A report submitted by BISHNU PADA MANDAL Roll No: 42MA2069 for the partial fulfilment for the award of the degree of Master of Science in Mathematics under the supervision of Dr. GOPAL KRISHNA PANDA DEPARTMENT OF MATHEMATICS NIT ROURKELA ROURKELA MAY 204

2 DECLARATION I hereby declare that the topic ON SUM AND RECIPROCAL SUMS OF FI- BONACCI NUMBERS for completion for my master degree has not been submitted in any other institution or university for the award of any other Degree, Fellowship or any other similar titles. Date: Place: Bishnu Pada Mandal Roll no: 42MA2069 Department of Mathematics NIT Rourkela i

3 CERTIFICATE This is to certify that the project report entitled ON SUM AND RECIPROCAL SUMS OF FIBONACCI NUMBERS submitted by Bishnu Pada Mandal to the National Institute of Technology Rourkela for the partial fulfilment of requirements for the degree of master of science in Mathematics is a bonafide record of review work carried out by him under my supervision and guidance. The contents of this project, in full or in parts, have not been submitted to any other institute or university for the award of any degree or diploma. May 204 Dr. Gopal Krishna Panda Professor Department of Mathematics NIT Rourkela ii

4 ACKNOWLEDGEMENT It is my pleasure to thank to the people, for whom this thesis is possible. I specially like to thanks my guide Dr. Gopal Krishna Panda, for his keen guidance and encouragement during the course of study and preparation of the final manuscript of this project. I am greatful to Prof. S.K.Sarangi, Director, National Institute of Technology Rourkela for providing excellent facilities in the institute for carrying out research. I would also like to thanks the faculty members of Department of Mathematics, National Institute of Technology Rourkela for his encouragement and valuable suggestion during the preparation of the thesis. I heartily thanks to my friends of NIT Rourkela who helps me for preparation of this project. I am also thankful to Mr. Ravi Kumar Davala and Mr. Akshaya Kumar Panda (Research Scholar, NIT Rourkela) for their true help and inspiration. Finally, thanks to my family members for their support and encouragement throughout my study and my career. Bishnu Pada Mandal iii

5 ABSTRACT The purpose of this report is to analyze the properties of Fibonacci numbers modulo a Lucas numbers. Any Fibonacci number, except the first two, is the sum of the two immediately preceding Fibonacci numbers and closely related to Fibonacci numbers are Lucas number. Fibonacci numbers are used in the application of computer algorithms. They can be used to compress audio files and generate code. The most recently Fibonacci number have been used to symbolize mathematical relationship in the Davinci code as well as in the TV shows fringe, criminal minds. In this report, some generalized identities of Holliday and Komatsu have been studied results of Liu and Zhao obtained by applying the floor function to the reciprocal of infinite sums of reciprocal generalized Fibonacci numbers and the infinite sums of reciprocal generalized Fibonacci sums have been extended. iv

6 Contents Introduction 2 Preliminaries 3 2. Lucas number Binet s formula Cassini s identity Pell numbers Associated Pell numbers Golden ratio Properties of Pell and Fibonacci Numbers 6 3. The simplest properties of Pell Numbers Some Properties of Fibonacci Numbers Binet s Formulae for Pell and Fibonacci Numbers Sums of Reciprocal Fibonacci Numbers 0 4. Introduction Some results related to the Reciprocals of generalized Fibonacci numbers. 0 5 Fibonacci Numbers Modulo a Lucas Number 3 5. Introduction Main Results Properties of Fibonacci Numbers Modulo m References 8 v

7 Chapter Introduction The Fibonacci numbers were first mentioned in 202 in the Liber Abaci, the book was written by Leonardo of Pisa to introduce the Hindu - Arabic numeral system to western Europe. They can be obtained by the recursive formula F n = F n + F n 2 F 0 = 0, F =, n 2 First derived from the famous rabbit problem of 228, the Fibonacci numbers were originally used to represent the number of pairs of rabbits born one pair in a certain population. Let us assume that a pair of rabbit is introduced in to a certain place in the first month of the year. This pair of rabbits take one month to became mature, and every pair of rabbits produces a mixed pair every month, from the second month and all rabbits are immortal.every pair of rabbits will produce perfectly on schedule. Suppose that the original pair of rabbits was born on January. They take a month to become mature, so there is still only one pair on February. On March, they are two months old and produce a new mixed pair, so total of two pairs. Continuing on, we find that we will have 3 pairs, in month April, 5 pairs in month May, then 8,3,2,34,... The Fibonacci sequence is one of the most famous and curious numerical sequence in the mathematics and have been widely studied from both algebraic and combinatorial prospectives. The Pell sequence {P n } are defined by recurrence. The Pell sequence, also obtained from the recurrence relation P n = 2P n + P n 2, P 0 = 0, P =, n 2 is also very important in number theory. The Diophantine equation x 2 dy 2 =, is known as the Pell s equation. Early mathematicians, upon discovering that 2 is irrational, realized the link of successive rational

8 approximations of 2 with x 2 dy 2 =. The early investigators of Pell s equation were the Indian mathematicians Brahmagupta and Bhaskara. In particular Bhaskara studied Pell s equation for the values d = 8,, 32, 6. and 67, Bhaskara found the solution x = , y = , for d = 6. Fermat was also interested in the Pell s equation and worked out some of the basic theories regarding Pell s equation. In general Pell s equation is a Diophantine equation of the form x 2 dy 2 =, where d is a positive non square integer and has a long fascinating history and its applications are wide and Pell s equation always has the trivial solution (x, y) = (, 0), and has infinite solutions and many problems can be solved using Pell s equation. 2

9 Chapter 2 2 Preliminaries 2. Lucas number Closely related to the Fibonacci numbers are called the Lucas numbers, 3, 4, 7,,... Lucas numbers are denoted by L n and recuresive relation as given below, L n = L n + L n 2, L =, L 2 = 3, n Binet s formula Both Fibonacci numbers and Lucas numbers can be defined explicity using Binet s formulas, F n = αn β n α β and L n = α n + β n where α = and β = 5 2 are the solution of the quadratic equation x 2 = x Cassini s identity Let {F n } be sequence of Fibonacci numbers, defined as F 0 =, F =, F n+ = F n + F n, n. Then for n, F n+ F n F 2 n = ( ) n+. 3

10 2.4 Pell numbers The Pell numbers can be obtained by the recurrence relation 0, if n = 0. P n =, if n =. 2P n + P n 2, otherwise. The Pell numbers sequence starts from 0 and and then each Pell number is the sum of twice the previous Pell number and the Pell number before that. 2.5 Associated Pell numbers The Associated Pell numbers can be obtained by the recurrence relation Q n+ = 2Q n + Q n, Q 0 =, Q =. 2.6 Golden ratio In Mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The golden ratio is also called the golden section or golden mean. It is denoted by φ and the approximately value is Two quantities a and b are said to be in the golden ratio φ if a + b a = a b = φ Then 4

11 a + b a = + a b = + φ + φ = φ φ 2 = φ + φ 2 φ = 0 φ = φ = φ.68 5

12 Chapter 3 3 Properties of Pell and Fibonacci Numbers 3. The simplest properties of Pell Numbers Theorem 3... Let a p = 2 p + Proof. Note that for all k > ( ) p 2p. Then a p < a p+ for p > p + 2 ( ) 2 k + < k k 2 k 2 Since, also k 2 > k 2 for all k >. Thus we can write for k > 2 2 ( ) 2 k + < k ( k 2 k 2 ) k Then we may write, ( ) 2 k + < 2 k ( k 2(k ) 2k ) k ( = (k + ) k 2(k ) ) k ( ) k 2k k + Therefore we get, k 2 k ( ) k 2(k ) < k ( ) k 2(k ) < k ( 2k k + k + 2 k + ( 2k k + ) ( 2k k + ) k ) k Which gives us for k > 2 a k < a k Thus, the proof is complete. Theorem The characteristic equation of the Pell numbers x p+ 2x p = 0 does not have multiple roots for p >. Proof. Let f(z) = z p+ 2z p. Suppose that α is a multiple root of f(z) = 0. Note that α 0 and α. Since α is a multiple root, f(α) = α p+ 2α p and f (α) = (p + )α p 2pα p = 0. Then f (α) = α p ((p + )α 2p) = 0 6

13 Thus, putting α = 2p p + and hence 0 = f(α) = α p+ + 2α p + = α p (2 α) + ( ) p ( 2p = 2 2p ) + p + p + ( ) p 2 2p = + p + p + = ap +. Since By above lemma, a 2 = > and a p < a p+ for p >, a p which is a contradiction. Therefore the equation f(z) = 0, does not have multiple roots. 3.2 Some Properties of Fibonacci Numbers Theorem F n 0(mod 2) if and only if n 0(mod 3). Proof. We prove by induction that for k N we have We have F 0 = 0 0(mod 2). We assume that F 3k 0(mod 2). F 3k 0(mod 2) Since, F k+l = F l F k+ + F l F k, we have F 3(k+) = F 3k+3 = F 3 F 3k+ + F 2 F 3k Since F 3 = 2 0(mod 2) and we assumed that F 3k 0(mod 2), we have We again prove by induction for k N F 3(k+) 0(mod 2) F 3k+ (mod 2) We have F = (mod 2). Let us assume that F 3k+ (mod 2). 7

14 From, F k+l = F l F k+ + F l F k, we have F 3(k+)+ = F 3k+4 = F 4 F 3k+ + F 3 F 3k Since F 3k 0(mod 2) and F 4 = 3 (mod 2), using the assumption F 3k+ (mod 2), it follows that F 3(k+)+ (mod 2) which completes the proof. Theorem F 5k 0(mod 5) with k N Proof. We again prove this result by induction. We have F 0 = 0 0(mod 5) Notice also that F 5 = 5 0(mod 5) Let assume that F 5k 0(mod 5) From, F k+l = F l F k+ + F l F k, we have F 5(k+) = F 5k+5 = F 5 F 5k+ + F 4 F 5k Since, F 5k 0(mod 5) and we assumed that F 5k 0(mod 5), we get Theorem F k+2 = + k i= F i Proof. We have F 5(k+) 0(mod 5). F 0 = 0 F = F 2 = F 0 + F F 3 = F + F 2 F 4 = F 2 + F 3 Continuing the above process, we get F k+2 = F k + F k+ 8

15 Adding these equalities, we get F k+2 = + F + F F k = + F k+2 = + k i= F i k i= F i 3.3 Binet s Formulae for Pell and Fibonacci Numbers Lemma The Binet s formula for the Pell sequence is Where γ = + 2, δ = 2. P n = γn δ n γ δ Lemma Let α = equation x 2 = x +. Then F n = αn β n 5, for all n. and β = 5, so that α and β are roots of the 2 Proof. When n =, F =, Which is true. Let us suppose that it is true for n =, 2, 3,...n. Then F k = αk β k and F k = αk β k. Adding these two equa- 5 5 tions, we get F k + F k = αk 5 ( + α ) + βk 5 ( + β ). Then F k+ = α(k+) + β (k+) 5. 9

16 Chapter 4 4 Sums of Reciprocal Fibonacci Numbers 4. Introduction Let a, b be two positive integers and c non negative integers. The generalized Fibonacci numbers F n (c; a, b) are defined by the following relation G 0 = c, G = and G n+ = ag n + bg n, (n ) Where F n (0;, ) = F n, are the Fibonacci numbers. F n (2;, ) = L n, are the Lucas numbers. F n (0; 2, ) = P n, are the Pell numbers. 4.2 Some results related to the Reciprocals of generalized Fibonacci numbers Theorem Let a,b be positive integers and c non negative integers. Then for n, we have () F n+ F n Fn 2 = ( ) n b n ( ac bc 2 ) (2) n i=0 F i = a+b n+ + bf n + ac c ) Theorem Let F n = V n (c;, ) for c, we have ( ) k i=0 F = F n (n n 0 ), i k=n Proof. By using the above lemma F n+ F n F 2 n = ( ) n ( c c 2 ) and n F n = F n+2 i=0 0

17 Suppose c, we have F n F n+ n i=0 F i = F n F n+ F n+2 F n+ = (F n )(F n+2 ) (F n+ ) = 2F n + ( ) n+ (c 2 + c ) (F n )(F n+ )(F n+2 ) Since F n is monotonic increasing with n, we can take n so large that 2F n ( ) n (c 2 + c) for a fixed c. Hence, the numerator of the right-hand side of the above identity is positive if n N for some positive integer N, So we get F n n i=0 F i + n i=0 F i + n i=0 F i + F n+ n+ i=0 F + i n+ i=0 F + i F n+2 n+2 i=0 F +... i Thus, On the other hand, we have k=n k i=0 F i F n (n N ) () n i=0 F + = i F n F n+ = F n+2 + F n F n+ F n (F n+2 ) + F n+ F n+ = F n + ( ) n (c 2 + c ) F n F n+ (F n+2 ) Similarly, we can take n so large that F n + ( ) n (c 2 + c ) > 0 for a fixed c. Hence, the numerator of the right-hand side of the above identity is positive if n N 2 for some positive integer N 2, we get F n < < < n i=0 F + i n i=0 F i + n i=0 F i + F n+ n+ i=0 F + i n+ i=0 F + i F n+2 n+2 i=0 F +... i

18 Thus, k=n k i=0 F i Combining the two inequalities () and (2), we get F n < k=n > F n (n N 2 ) (2) k i=0 F i F n Where n n 0 = max{n, N 2 }, which completes the proof. Theorem Let a b and F n = V n (0; a, b), we have ( ) k i=0 F = F n (n N a ), i k=n Where N a = 3, for a = and N a = 2, for a 2 Proof. The case a =, has already been proved, it is sufficient to show the case a 2. Defining S n = n i=0 F i, we have S n F n + F n+ = F n+ S n F n S n = F ns n F n+ S n F n F n+ S n = F n+ F n + ( ) n+ af n F n+ S n > 0 and for n 2 F n F n+ = S n + S n (F n )S n F n+ = F n+s n F n S n + as n (F n )(F n+ )S n 2F n + ( ) n = a(f n )(F n+ )S n > 0 Then, we get which completes the proof. F n < k=n (n 2) S k F n 2

19 Chapter 5 5 Fibonacci Numbers Modulo a Lucas Number 5. Introduction Let m, m 2,... be positive integers, then k := sup min xm i. x (0,) i Here, for x R, x is the distance to the nearest integer. Observe that if we have a finite number of integers m, m 2,...m n and k := max m=m j +m l m min km i m i Where x m denotes the absolute value of the absolutely least remainder of x mod m. 5.2 Main Results Let M = {F 2, F 3,...F t }. Let n be an integer such that 4k + 2 t 4k + 5 and m = F 2k+2 + F 2k+4 = L 2k+3. In this section, we use the following identities.. Cassini s identity: F k+ F k F 2 k = ( )k+. 2. F 2 k + F 2 k+ = F 2k+ 3. d Ocagne s identity: F m F k+ F m+ F k = ( ) k F m k 4. F 2k = k i=0 F 2i+ 5. F 2k+ = k i= F 2i 3

20 5.3 Properties of Fibonacci Numbers Modulo m Lemma (a) F 2 F 2k+2 F 4k+5 F 2k+2 (mod m) (b) F 2 F 2k+2 F 4k+4 F 2k+2 (mod m) (c) F 3 F 2k+2 F 4k+3 F 2k+2 (mod m) (d) F 4 F 2k+2 F 4k+2 F 2k+2 (mod m) Proof. (a) We have mf 2k+2 = (F 2k+2 + F 2k+4 )F 2k+2 = F2k F 2k+4 F 2k+2 = F2k (F 2k+3 + F 2k+2 )(F 2k+3 F 2k+ ) = F2k F2k F 2k+3 F 2k+2 F 2k+3 F 2k+ F 2k+2 F 2k+ = F2k F2k (F 2k+2 + F 2k+ )F2k+2 2 F 2k+3 F 2k+ F 2k+2 F 2k+ = F2k F2k F2k+2 2 F 2k+3 )F 2k+ = F2k F2k ( ) 2k+ [Cassini s identity] = F 4k+5 Hence, we get F 4k+5 F 2k+2 = ( + mf 2k+2 )F 2k+2 F 2k+2 = F 2 F 2k+2 (mod m) Proof. (b) By using (a) i.e F 2 F 2k+2 F 4k+5 F 2k+2 (mod m) F 2 F 2k+2 F 4k+5 F 2k+2 (mod m) = (F 4k+6 F 4k+4 )F 2k+2 = F 4k+6 F 2k+2 F 4k+4 F 2k+2 = (L 2k+3 F 2k+3 )F 2k+2 F 4k+4 F 2k+2 [Since L k F k = F 2k ] F 4k+4 F 2k+2 (modm) 4

21 Proof. (c) i.e F 2 F 2k+2 F 4k+4 F 2k+2 (mod m) F 3 F 2k+2 = (F 2 + F )F 2k+2 = F 2 F 2k+2 + F F 2k+2 = F 4k+5 F 2k+2 + F 2 F 2k+2 [By using (a) and F = F 2 ] = F 4k+5 F 2k+2 F 4k+4 F 2k+2 [By using (b)] (F 4k+5 F 4k+4 )F 2k+2 F 4k+3 F 2k+2 (mod m) Proof. (d) i.e F 3 F 2k+2 F 4k+3 F 2k+2 (mod m) F 4 F 2k+2 = (F 3 + F 2 )F 2k+2 = F 3 F 2k+2 + F 2 F 2k+2 (F 4k+3 F 4k+4 )F 2k+2 [By using (b) and (c) ] F 4k+2 F 2k+2 (modm) i.e F 4 F 2k+2 F 4k+2 F 2k+2 (mod m) Lemma F p+5 F 2k+2 = (F p+4 + F p+3 )F 2k+2 ɛf 4k+ p F 2k+2 (mod m) for each 0 p 2k 2, where +, if p is even. ɛ =, if p is odd. (3) Proof. Using the recurrence relation F K+ = F k + F k for k and Lemma 4.3., the proof follows by induction on p. Lemma (a) F 2 F 2k+2 F 2k+2 (mod m). (b) F 3 F 2k+2 F 2k+3 (mod m). (c) F 4 F 2k+2 F 2k+ (mod m). (d) F 5 F 2k+2 2F 2k+2 F 2k+ (mod m). 5

22 Proof. (a) F 2 F 2k+2 = F 2k+2 F 2k+2 (mod m) Proof. (b) i.e F 2 F 2k+2 F 2k+2 (mod m). F 3 F 2k+2 = 2F 2k+2 (As F 3 = 2) (F 2k+ + F 2k+2 ) = F 2k+3 (mod m) Proof. (c) i.e F 3 F 2k+2 F 2k+3 (mod m). F 4 F 2k+2 = (F 2 + F 3 )F 2k+2 = F 2 F 2k+2 + F 3 F 2k+2 (F 2k+2 F 2k+3 ) [F 2 = and using (b)] F 2k+ (mod m) Proof. (d) ı.e F 4 F 2k+2 F 2k+ (mod m). F 5 F 2k+2 = 5F 2k+2 (AsF 5 = 5) = (F 3 + F 4 )F 2k+2 [using (b) and (c)] = F 3 F 2k+2 + F 4 F 2k+2 2F 2k+2 F 2k+ (mod m) 6

23 i.e F 5 F 2k+2 2F 2k+2 F 2k+ (mod m). 7

24 References [] D.M Burton; Elementary Number Theory; Tata Mc Graw Hill. [2] Thomas Koshy; Elementary Number Theory with Application; Elsevier. [3] K. Kuhapatanakul. On the Sums of Reciprocal Generalized Fibonacci Numbers. Journal of Integer Sequences.Vol.6, 203 [4] R. K. Pandey. On Some Magnified Fibonacci Numbers Modulo a Lucas Number. Journal of Integer Sequences.Vol.6, 203 [5] Hardy G.H., Wright; An Introduction to the Theory of Numbers; Oxford Science Publication, Fifth Edition (979). [6] E. Klc. The generalized Pell (p,i)-numbers and their Binet formulas,combinatorial representations, sums Elsevier, [7] A. Laugier and M. P. Saikia. Some Properties Of Fibonacci Numbers.arXiv: v3 [math.nt] Jul 203 [8] S.H.Holliday and T.Komatsu, On the sum of reciprocal generalized Fibonacci Numbers, Integers A (20). [9] H.Ohtsuka and S.Nakamura, On the sum of reciprocal sums of Fibonacci numbers, Fibonacci Quart. 46/47 (2008/2009), [0] Z.Wenpeng W.Tingting, The infinite sum of reciprocal Pell numbers, Appl. Math. Comput. 28 (202), [] W.Bienia, L.Goddyn, P.Gvozdjak, A.Sebo, and M.Tarsi, Flows, view-obstructions and the lonely runner, J.Combin. Theory Ser. B 72 (998), -9. [2] T.W.Cusick,View-obstruction problems in n-dimensional geometry, J.Combin. Theory ser. A 6 (974), -. [3] N.M.Haralambis, Sets of integers with missing differences, J.Cumbin. Theory Ser. A 23(977),

25 [4] T.Koshy, Fibonacci and Lucas Numbers with Applications, Willy, 200. [5] R.Liu and F.Z. Zhao, On the sums of reciprocal hyperfibonacci numbers and hyperlucas numbers, J. Integer Seq. 5 (202), Article