THE FIBONACCI SEQUENCE MODULO M

Introduction

Here are some of the main results which are known about the Fibonacci sequence under a modulus. Also, the section on Fibonacci subsequences is an area where I have done original research. I do not prove all the statements here, but only the ones which are fairly short and interesting, in the hope that the document will have a nice flow.

Thanks for dropping by and I hope that you enjoy reading about these amazing properties of the Fibonacci sequence!

THE PERIOD OF F(MOD M)

The Basics

1. Definition: By F(mod m) we mean the sequence of least nonnegative residues of the terms of the Fibonacci sequence taken modulo m. This is a bi-infinite sequence in that given any two consecutive terms, we can find the terms preceeding and following those terms. For example, I am using F(mod 5) as the horizontal rule above.

2. F(mod m) is periodic. This is a natural consequence of the following two statements:
   • Modulo m, there are m² possible pairs of residues, and hence some pair of consecutive terms of F(mod m) must repeat.
   • Any pair of consecutive terms of F(mod m) completely determines the entire sequence.

3. Definition: Let k(m) denote the period of F(mod m). Let k = k(m) and note the two easy consequences:
   • F_k = 0 and F_k+1 = 1 mod m.
   • If F_n = 0 and If F_n+1 = 1 mod m, then k|n.

4. Given any integer m, infinitely many Fibonacci numbers are divisible by m.
   Proof:
   • The pair 0,1 is in the Fibonacci sequence, and so, for m > 1, it will appear in F(mod m).
   • Since F(mod m) is periodic, this means that the pair 0,1 will appear infinitely many times in F(mod m). This proves our result.

5. The zeros of F(mod m) are evenly spaced.
   • From identities 1 and 2 at the end of this document, it follows that if F_s = F_t = 0 mod m, then F_s+t = 0 mod m.

Goal: Express the period of F(mod m) in terms of m.

1. For m > 2, k(m) is even.
   • Let k = k(m) and consider all equalities to be congruences modulo m. We will prove the contrapositive of the above statement.
   • Since F_k = 0 and F_k+1 = 1, we see F_k-1 = 1. Similarly, F_-k+1 = 1. Now assume that k is odd. Then by identity 3 at the end of this document, F_k-1 = -F_-(k-1) = -1. Since 1 = -1 mod m, we conclude that m = 2.

2. If n | m, then k(n) | k(m).
   • For example, 17 | 51, and k(17) = 36 and k(51) = 72. Interestingly, though 51/17 = 3, k(51)/k(17) = 2.

3. Let m be factored into primes, p_i^e_i. Then k(m) = lcm[k(p_i^e_i)].
   • By statement (2) above, we see k(p_i^e_i) | k(m) for all i.
     Thus lcm[k(p_i^e_i)] | k(m).
   • Let L = lcm[k(p_i^e_i)]. Since k(p_i^e_i) | L for each i, the sequence F(mod p_i^e_i) repeats in blocks of length L (and maybe less) for each i.
     This implies F_L = 0 and F_L+1 = 1 (mod p_i^e_i) for all i.
     Hence F_L = 0 and F_L+1 = 1 (mod m) since the p_i^e_i are relatively prime.
     Thus k(m) | L.
   For example, 90 = (2)(3²)(5).
   k(2) = 3, k(9) = 24, k(5) = 20.
   lcm[3, 24, 20] = 120 = k(90).

This last fact tells us that if we can find k(p^e) for primes p, then we can find k(m) for any m, which is our ultimate goal.

Revised Goal: Express k(p^e) in terms of p^e.

1. Given a prime p, let t be the largest integer such that k(p^t) = k(p). Then k(p^e) = p^e-tk(p) for all e >= t.
   • For example, k(3¹) = 8, but k(3²) = 16. Thus for the prime 3, t = 1. Now we can find k(3^e) for any e we desire. For example, k(3⁴) = 3^4-1k(3) = (27)(8) = 216.

2. Conjecture: t = 1 for all primes p. That is, the above theorem can be restated: k(p^e) = p^e-1k(p).

Now, all we have to do is determine k(p) in order to find k(p^e) in order to find k(m), which is our ultimate goal.

Revised Revised Goal: Express k(p) in terms of p.

Unfortunately, at this point we get stuck. There are no explicit formulas for k(p) in terms of p, but we can find some bounds on the value of k(p).
1. Upper bound on k(p).
   1. If p = 5n ± 1, then k(p) | (p-1).
      • For example, k(71) = 70 and k(89) = 44.
   2. If p = 5n ± 2, then k(p) | (2p+2).
      • For example, k(73) = 148 and k(47) = 32.

2. Lower bound on k(m): Given a modulus m, let t > 0 be chosen so that L_t is >= m, where L_t is the t^th Lucas number. Then k(m) >= 2t.
   • Suppose we wish to find a lower bound for k(987). The 14th Lucas number, L₁₄ = 843. So by the above theorem, k(987) >= 28. In fact, k(987) = 32.
     However, this lower bound is not always so precise: k(985) = 1980.

THE ZEROS OF F(MOD M)

Preliminaries

1. Definition: Let a(m) denote the index of the first Fibonacci number which is divisible by m. We call a(m) the "restricted period" of F(mod m). Equivalently, it is the position of the first zero in the sequence F(mod m).

2. Definition: Let s(m) denote first residue appearing after the first zero in F(mod m). We call s(m) the "mulitplier" of F(mod m).

3. Definition: Let b(m) denote the order of s(m), modulo m, if it exists. (We'll see it always exists.)

4. Example: Consider F(mod 5): 0 1 1 2 3 0 3 3 1 4 0 4 4 3 2 0 2 2 4 1 0 1 1 ...
   The period, k(5) = 20.
   The restricted period, a(5) = 5.
   The multiplier is s(5) = 3.
   The order of 3, mod 5, i.e. b(5), is 4 since 3⁴ = 1 (mod 5), but 3¹, 3², 3³ <> 1 (mod 5).

5. a(m)b(m) = k(m).
   • Let a = a(m), b = b(m), k = k(m), s = s(m).
     Let G_n denote the sequence F(mod m) except that the starting point of G_n is the n^th term of F(mod m). For example, G₀ starts off 0 1 1 ... whereas G_a begins 0 s s ...
     Essentially, G_a is really just G₀ with every term multiplied by s. We can write this as G_a = (s)G₀.
     Similarly, G_2a = (s)G_a = (s²)G₀.
     Eventually we get G_ba = (s^b)G₀ = G₀ since b is the order of s.
     Also, since b is the order of s, it follows that ab = k.

   • Hence we have another interpretation for b(m): it is the number of zeros in one period of F(mod m).

Properties of b(m)

1. b(m) = 1, 2, or 4.
   • This is an amazing result. No matter how large our modulus is, every period of F(mod m) will have either 1, 2, or 4 zeros. We present the proof next which is very interesting.

   • By identity 4 at the end of this document, F_t² - F_t+1F_t-1 = (-1)^t+1.
     So in particular, F_a² - F_a+1F_a-1 = (-1)^a+1.
     Since F_a = 0 and F_a+1 = F_a-1 = s, we get -s² = (-1)^a+1.
     That is, s² = (-1)^a.
     Since these two quantities are congruent modulo m, they have the same order. We know that the order of s is b. Let us denote the order of -1 by c. Hence c = 1 if m = 2, and c = 2 otherwise. Now we equate orders of s² and (-1)^a:
     b / gcd(2, b) = c / gcd(a, c).
     k = ab = a( (c)gcd(2, b) / gcd(a, c) ).
     So k = lcm[a, c]gcd(2, b).
     So k = (a or 2a)(1 or 2).
     So k = a or 2a or 4a.
     In other words, b = 1, 2, or 4.

2. If p is an odd prime, then b(p^e) = b(p).

3. For m > 2, b(m) = 4 iff a(m) is odd.

4. b(m) = 1 iff 4 does not divide k(m).

5. b(m) = 2 iff 4 | k(m) and a(m) is even.

6. If 3 | m, then b(m) = 2.

FIBONACCI SUBSEQUENCES

Example and Definition

1. Example: Consider the sequence F(mod 6) = 0 1 1 2 3 5 2 1 3 4 1 5 0 5 5 4 3 1 4 5 3 2 5 1 0 1 1 ... If we take every third term, we have a new sequence: 0 2 2 4 0 4 4 2 0 2 2 ... If we take every fourth term, we get 0 3 3 0 3 3 0 3 3 0 3 ... And, if we take every 11th term we have the sequence 0 5 5 4 3 1 4 5 3 2 5 ... which is just our original sequence with a new starting point.

2. Definition: All of the above subsequences of F(mod 6) also exhibit the Fibonacci recurrence relation, mod 6. We call such sequences Fibonacci subsequences.

Results

1. If F_z = 0, and F_z+n = F_z+2n, then F_z, F_z+n, F_z+2n, F_z+3n, ... forms a Fibonacci subsequence.

2. Conjecture: If m is not a multiple of 5, then all Fibonacci subsequences of F(mod m) must contain a zero. Thus, all Fibonacci subsequences of F(mod m) can be found by using the requirements in the above point.

3. Conjecture: If any four terms of a subsequence of F(mod m) exhibit the Fibonacci recurrence relation, then that sequence is a Fibonacci subsequence.

Open Questions

1. Given m, how many Fibonacci subsequences of F(mod m) are there?

2. How can we classify and describe these Fibonacci subsequences? In the example at the start of the Fibonacci subsequences section we can say that the three example subsequences are 2F(mod 3), 3F(mod 2), and F(mod 6). Given m, what kind of subsequences might appear?

3. Just about any other question you can ask. There is very little literature in this area, and yet there seem to be some very interesting relationships going on here.

Identities

2. F_s-t = (-1)^t (F_sF_t+1 - F_s+1F_t)

3. F_-t = (-1)^t+1 F_t

4. F_t² - F_t+1F_t-1 = (-1)^t+1

Marc Renault