Divisibility in the Fibonacci Numbers. Stefan Erickson Colorado College January 27, 2006

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1 Divisibility in the Fibonacci Numbers Stefan Erickson Colorado College January 27, 2006

2 Fibonacci Numbers F n+2 = F n+1 + F n n F n n F n n F n n F n

3 Lucas Numbers L n+2 = L n+1 + L n n L n n L n n L n n L n

4 Fundamental Questions Is there a formula for the n th Fibonacci number? Which Fibonacci numbers divide other Fibonacci numbers? Which primes divide the Fibonacci numbers? What is the entry point for each prime? What about the Lucas numbers?

5 Guess: F n = r n for some r. Recurrence Relations F n+2 = F n+1 + F n

6 Recurrence Relations Guess: F n = r n for some r. F n+2 = F n+1 + F n r n+2 = r n+1 + r n r n (r 2 r 1)=0

7 Recurrence Relations Guess: F n = r n for some r. F n+2 = F n+1 + F n r n+2 = r n+1 + r n r n (r 2 r 1)=0 r 1 =φ= 1+ 2 r 2 =φ = 1 2 = =

8 Recurrence Relations Guess: F n = r n for some r. F n+2 = F n+1 + F n r n+2 = r n+1 + r n r n (r 2 r 1)=0 r 1 =φ= 1+ 2 r 2 =φ = 1 2 = = φ φ = 1 φ n =F n φ+f n 1

9 Binet s Formula General Solution: F n = c 1 r n 1 + c 2 r n 2

10 Binet s Formula General Solution: F n = c 1 r n 1 + c 2 r n 2 F 0 =0,F 1 =1 = c 1 = 1,c 2 = 1 F n = 1 ( φ n φ n)

11 Binet s Formula General Solution: F n = c 1 r n 1 + c 2 r n 2 F 0 =0,F 1 =1 = c 1 = 1,c 2 = 1 F n = 1 ( φ n φ n) L 0 =2,L 1 =1 = c 1 =1,c 2 =1 L n =φ n +φ n

12 Generating Functions F n x n = n=0 x 1 x x 2

13 Generating Functions F n x n = n=0 = x 1 x x 2 x (1 φx)(1 φ x)

14 F n x n = n=0 Generating Functions = x 1 x x 2 x (1 φx)(1 φ x) = φx φ x

15 F n x n = n=0 Generating Functions x 1 x x 2 x = (1 φx)(1 φ x) = φx φ x = 1 n=0 (φx) n 1 n=0 (φ x) n

16 F n x n = n=0 Generating Functions x 1 x x 2 x = (1 φx)(1 φ x) = φx φ x = 1 = n=0 n=0 (φx) n 1 1 ( φ n φ n) x n n=0 (φ x) n

17 Consequences lim n F n+1 F n = lim n L n+1 L n = φ

18 Consequences lim n F n+1 F n = lim n L n+1 L n = φ lim n L n F n =

19 Consequences lim n F n+1 F n = lim n L n+1 L n = φ lim n L n F n = F 2n = F n L n

20 Consequences lim n F n+1 F n = lim n L n+1 L n = φ lim n L n F n = ( Proof. F 2n = F n L n 1 ( φ 2n φ 2n) = 1 ( φ n φ n) (φ n + φ n))

21 F n = F n 1 + F n 2 Super Cool Formula

22 Super Cool Formula F n = F n 1 + F n 2 =(F n 2 +F n 3 )+F n 2 =2F n 2 +1F n 3

23 Super Cool Formula F n = F n 1 + F n 2 =(F n 2 +F n 3 )+F n 2 =2(F n 3 +F n 4 )+F n 3 =2F n 2 +1F n 3 =3F n 3 +2F n 4

24 Super Cool Formula F n = F n 1 + F n 2 =(F n 2 +F n 3 )+F n 2 =2(F n 3 +F n 4 )+F n 3 =2F n 2 +1F n 3 =3F n 3 +2F n 4 =3(F n 4 +F n )+2F n 4 =F n 4 +3F n

25 Super Cool Formula F n = F n 1 + F n 2 =(F n 2 +F n 3 )+F n 2 =2(F n 3 +F n 4 )+F n 3 =2F n 2 +1F n 3 =3F n 3 +2F n 4 =3(F n 4 +F n )+2F n 4 =F n 4 +3F n F m+n =F m F n+1 + F m 1 F n L m+n = F m L n+1 + F m 1 L n

26 Theorem. F n divides F kn for all integers k.

27 Theorem. F n divides F kn for all integers k. Proof. F n divides F n, so the theorem is true for k =1. Suppose F n divides F kn. Then F (k+1)n = F kn F n+1 + F kn 1 F n is also divisible by F n. By induction, the theorem is true for all k!

28 Theorem. F n divides F kn for all integers k. Proof. F n divides F n, so the theorem is true for k =1. Suppose F n divides F kn. Then F (k+1)n = F kn F n+1 + F kn 1 F n is also divisible by F n. By induction, the theorem is true for all k! Theorem. L n divides L kn for all odd integers k.

29 Theorem. F n divides F kn for all integers k. Proof. F n divides F n, so the theorem is true for k =1. Suppose F n divides F kn. Then F (k+1)n = F kn F n+1 + F kn 1 F n is also divisible by F n. By induction, the theorem is true for all k! Theorem. L n divides L kn for all odd integers k. Proof. Same proof, except L (2l+1)n = F 2ln L n+1 + F 2ln 1 L n and L n divides F 2n, which in turn divides F 2ln.

30 Greatest Common Divisors Theorem. gcd(f m,f n )=F gcd(m,n). Ex: gcd(144, 284) = 8.

31 Greatest Common Divisors Theorem. gcd(f m,f n )=F gcd(m,n). Ex: gcd(144, 284) = 8. Proof. Let d = gcd(m, n). By the previous theorem, F d divides F m and F n. Thus, F d divides gcd(f m,f n ).

32 Greatest Common Divisors Theorem. gcd(f m,f n )=F gcd(m,n). Ex: gcd(144, 284) = 8. Proof. Let d = gcd(m, n). By the previous theorem, F d divides F m and F n. Thus, F d divides gcd(f m,f n ). d may be written as am + bn for some integers a and b. Then F d = F am+bn = F am F bn+1 + F am 1 F bn Since F m divides F am and F n divides F bn, F d can be written as a linear combination of F m and F n. Hence, gcd(f m,f n ) divides F d.

33 Question: Which Fibonacci numbers divide other Fibonacci numbers? Answer: F n divides F m if and only if n divides m.

34 Question: Which Fibonacci numbers divide other Fibonacci numbers? Answer: F n divides F m if and only if n divides m. For example, the Fibonacci numbers which divide F 30 = are F 3 =2,F =,F 6 =8,F 10 =, and F 1 = 610.

35 Question: Which Fibonacci numbers divide other Fibonacci numbers? Answer: F n divides F m if and only if n divides m. For example, the Fibonacci numbers which divide F 30 = are F 3 =2,F =,F 6 =8,F 10 =, and F 1 = 610. In order for F n to be prime, n must be prime. The converse is not true, since F 19 =4181=

36 Prime Divisibility Question: Which primes divide the Fibonacci numbers?

37 Prime Divisibility Question: Which primes divide the Fibonacci numbers? Answer: All of them! If p, then p divides either F p 1 or F p+1.

38 Prime Divisibility Question: Which primes divide the Fibonacci numbers? Answer: All of them! If p, then p divides either F p 1 or F p+1. For example, 7 divides F 8 = 21 and 11 divides F 10 =. Note that F = is the only exception.

39 Lemma. φ p φ (mod p) if p 1, 4 (mod ) φ (mod p) if p 2, 3 (mod )

40 Lemma. φ p φ (mod p) if p 1, 4 (mod ) φ (mod p) if p 2, 3 (mod ) Proof. φ p = 1 2 p ( 1+ ) p

41 Lemma. φ p φ (mod p) if p 1, 4 (mod ) φ (mod p) if p 2, 3 (mod ) Proof. φ p = 1 2 p ( 1+ ) p = 1 ( 1+ ( ) ( p + + p 2 p 1 p 1 )( ) p 1 + ( ) p )

42 Lemma. φ p φ (mod p) if p 1, 4 (mod ) φ (mod p) if p 2, 3 (mod ) Proof. φ p = 1 2 p ( 1+ ) p = 1 ( 1+ ( ) ( p + + p 2 p (1+ p 1 2 ) p 1 )( ) p 1 + ( ) p ) (mod p)

43 Lemma. φ p φ (mod p) if p 1, 4 (mod ) φ (mod p) if p 2, 3 (mod ) Proof. φ p = 1 2 p ( 1+ ) p = 1 ( 1+ ( ) ( p + + p 2 p (1+ p 1 2 ) 1 2 p 1 ( 1+ ( )( ) p 1 + ( ) p ) p ) ) (mod p)

44 Lemma. φ p φ (mod p) if p 1, 4 (mod ) φ (mod p) if p 2, 3 (mod ) Proof. φ p = 1 2 p ( 1+ ) p = 1 ( 1+ ( ) ( p + + p 2 p (1+ p 1 2 ) 1 2 p 1 ( 1+ ( )( ) p 1 + ( ) p ) p ) ) (mod p) ( ) p =1ifp 1,4 (mod ) and ( p ) = 1 ifp 2,3 (mod ).

45 Theorem. p divides F p 1 if p 1, 4 (mod ). (Ex: 11 F 10 =.) p divides F p+1 if p 2, 3 (mod ). (Ex: 7 F 8 = 21.)

46 Theorem. p divides F p 1 if p 1, 4 (mod ). (Ex: 11 F 10 =.) p divides F p+1 if p 2, 3 (mod ). (Ex: 7 F 8 = 21.) Proof. If p 1, 4 (mod ), then φ p φ (mod p). Dividing by φ, φ p 1 1 (mod p). F p 1 = 1 ( φ p 1 φ p 1) 1 ( 1 1 ) 0 (mod p).

47 Theorem. p divides F p 1 if p 1, 4 (mod ). (Ex: 11 F 10 =.) p divides F p+1 if p 2, 3 (mod ). (Ex: 7 F 8 = 21.) Proof. If p 1, 4 (mod ), then φ p φ (mod p). Dividing by φ, φ p 1 1 (mod p). F p 1 = 1 ( φ p 1 φ p 1) 1 ( 1 1 ) 0 (mod p). If p 2, 3 (mod ), then φ p φ (mod p). Multiplying by φ, φ p+1 1 (mod p). F p+1 = 1 ( φ p+1 φ p+1) 1 ( 1+1 ) 0 (mod p).

48 Theorem. If p k divides F n for some k 1, then p k+1 divides F pn.

49 Theorem. If p k divides F n for some k 1, then p k+1 divides F pn. Proof. Homework problem.

50 Theorem. If p k divides F n for some k 1, then p k+1 divides F pn. Proof. Homework problem. Corollary. Every positive integer divides F n for some n. For example, 1000 = 8 12 divides F 6 12 = F 70.

51 Entry Points The smallest n such that p divides F n is called the entry point of p. We use e F (p) for the entry point of p into the Fibonacci numbers and e L (p) for the entry point of p into the Lucas numbers.

52 Entry Points The smallest n such that p divides F n is called the entry point of p. We use e F (p) for the entry point of p into the Fibonacci numbers and e L (p) for the entry point of p into the Lucas numbers. From the previous theorem, e F (p) divides either p 1orp+1. For example, we know that 13 divides the 14 th Fibonacci number. However, 13 also divides F 7 = 13, so e F (13) = 7. On the other hand, 13 does not divide any Lucas number. We say that e L (13) is undefined in this case.

53 Entry Points for the Lucas Numbers If e F (p) is even, then e L (p) = e F(p) 2. If e F (p) is odd, then p does not divide any Lucas number. This follows from F 2n = F n L n. For example, e F (29) = 14 and e L (29) = 7. Question: Which primes divide the Lucas numbers?

54 One More Formula L 2 n F 2 n =4 ( 1) n

55 One More Formula L 2 n F 2 n =4 ( 1) n Lemma. If p does not divide L n for any n, then p 1 (mod 4). Proof. Suppose n = e F (p) is odd, that is, p F n for some odd n. L 2 n 4 (mod p) = 1 is a square mod p = p 1 (mod 4).

56 One More Formula L 2 n F 2 n =4 ( 1) n Lemma. If p does not divide L n for any n, then p 1 (mod 4). Proof. Suppose n = e F (p) is odd, that is, p F n for some odd n. L 2 n 4 (mod p) = 1 is a square mod p = p 1 (mod 4). Corollary. All primes p 3 (mod 4) divide the Lucas numbers.

57 L 2 n F 2 n =4 ( 1) n Lemma. If p 1 (mod 4) and p 2, 3 (mod ), then p does not divide L n for any n.

58 L 2 n F 2 n =4 ( 1) n Lemma. If p 1 (mod 4) and p 2, 3 (mod ), then p does not divide L n for any n. Proof. Suppose p L n. Then F 2 n 4 ( 1) n (mod p) = ± is a square mod p. By quadratic reciprocity, ± are not squares when p 1 (mod 4) and p 2, 3 (mod ).

59 p 1 (mod 4) p 3 (mod 4) p 1, 4 (mod ) p 2, 3 (mod )

60 p 1 (mod 4) p 3 (mod 4) p 1, 4 (mod ) p L n for some n p 2, 3 (mod ) p L n for some n

61 p 1 (mod 4) p 3 (mod 4) p 1, 4 (mod ) p L n for some n p 2, 3 (mod ) p L n for any n p L n for some n

62 p 1 (mod 4) p 3 (mod 4) p 1, 4 (mod )? p L n for some n p 2, 3 (mod ) p L n for any n p L n for some n

63 Suppose p 1 (mod 4) and p 1, 4 (mod ) such that p L n for some n. We may assume n p 1 2. Then L n = φ n + φ n 0 (mod p) φ n φ n (mod p) φ 2n ( 1) n+1 (mod p)

64 φ 2n ( 1) n+1 (mod p) If n = p 1 4 is odd, then φ p (mod p) means φ is a square mod p. Hence, 1 2 of the primes p (mod 8) divide L p 1. 4 If n = p 1 4 is even, then φ p (mod p) means φ is not a square mod p. Hence, 1 2 of the primes p 1 (mod 8) divide L p 1. 4

65 φ 2n ( 1) n+1 (mod p) If n = p 1 8 is odd, then φ p (mod p) means φ is a fourth power mod p. Hence, 1 4 of the primes p 9 (mod 16) divide L p 1. 8 If n = p 1 8 is even, then φ p (mod p) means φ is a square but not a fourth power mod p. Hence, 1 4 of the primes p 1 (mod 16) divide L p 1. 8

66 Question: How many primes divide the Lucas numbers? Answer: = 1 2 ( 1 4 n=0 ) n = 2 3 The density of the primes dividing the Lucas numbers is 2 3!

67 Conclusions F n divides F m if and only if n divides m.

68 Conclusions F n divides F m if and only if n divides m. L n divides L m if and only if m is an odd multiple of n.

69 Conclusions F n divides F m if and only if n divides m. L n divides L m if and only if m is an odd multiple of n. Every positive integer divides the Fibonacci numbers.

70 Conclusions F n divides F m if and only if n divides m. L n divides L m if and only if m is an odd multiple of n. Every positive integer divides the Fibonacci numbers. 2 3 of the primes divide the Lucas numbers.

71 Conclusions F n divides F m if and only if n divides m. L n divides L m if and only if m is an odd multiple of n. Every positive integer divides the Fibonacci numbers. 2 3 of the primes divide the Lucas numbers. The Fibonacci and Lucas numbers are cool!

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