Pisano period
In number theory, the nth Pisano period, written π(n), is the period with which the sequence of Fibonacci numbers taken modulo n repeats. Pisano periods are named after Leonardo Pisano, better known as Fibonacci. The existence of periodic functions in Fibonacci numbers was noted by Joseph Louis Lagrange in 1774.[1][2]
Definition
The Fibonacci numbers are the numbers in the integer sequence:
- 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, ... (sequence A000045 in the OEIS)
defined by the recurrence relation
For any integer n, the sequence of Fibonacci numbers Fi taken modulo n is periodic. The Pisano period, denoted π(n), is the length of the period of this sequence. For example, the sequence of Fibonacci numbers modulo 3 begins:
- 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, ... (sequence A082115 in the OEIS)
This sequence has period 8, so π(3) = 8.
Properties
With the exception of π(2) = 3, the Pisano period π(n) is always even. A simple proof of this can be given by observing that π(n) is equal to the order of the Fibonacci matrix
in the general linear group GL2(ℤn) of invertible 2 by 2 matrices in the finite ring ℤn of integers modulo n. Since F has determinant -1, the determinant of Fπ(n) is (-1)π(n), and since this must = 1 in ℤn, either n≤2 or π(n) is even.[3]
If m and n are coprime, then π(mn) is the least common multiple of π(m) and π(n), by the Chinese remainder theorem. For example, π(3) = 8 and π(4) = 6 imply π(12) = 24. Thus the study of Pisano periods may be reduced to that of Pisano periods of prime powers q = pk, for k ≥ 1.
If p is prime, π(pk) divides pk–1 π(p). It is conjectured that for every prime p and integer k > 1. Any prime p providing a counterexample would necessarily be a Wall-Sun-Sun prime, and such primes are also conjectured not to exist.[4]
So the study of Pisano periods may be further reduced to that of Pisano periods of primes. In this regard, two primes are anomalous. The prime 2 has an odd Pisano period, and the prime 5 has period that is relatively much larger than the Pisano period of any other prime. The periods of powers of these primes are as follows:
- If n = 2k, then π(n) = 3·2k–1 = 3·2k/2 = 3n/2.
- if n = 5k, then π(n) = 20·5k–1 = 20·5k/5 = 4n.
From these it follows that if n = 2·5k then π(n) = 6n.
The remaining primes all lie in the conjugacy classes or . If p is a prime different from 2 and 5, then the modulo p analogue of Binet's formula implies that π(p) is the multiplicative order of the roots of x2 – x – 1 modulo p. If , these roots belong to (by quadratic reciprocity). Thus their order, π(p) is a divisor of p – 1. For example, π(11) = 11 – 1 = 10 and π(29) = (29 – 1)/2 = 14.
If the roots modulo p of x2 – x – 1 do not belong to (by quadratic reciprocity again), and belong to the finite field
As the Frobenius automorphism exchanges these roots, it follows that, denoting them by r and s, we have rp = s, and thus rp+1 = –1. That is r2(p+1) = 1, and the Pisano period, which is the order of r, is the quotient of 2(p+1) by an odd divisor. This quotient is always a multiple of 4. The first examples of such a p, for which π(p) is smaller than 2(p+1), are π(47) = 2(47 + 1)/3 = 32, π(107) = 2(107 + 1)/3 = 72 and π(113) = 2(113 + 1)/3 = 76. (See the table below)
It follows from above results, that if n = pk is an odd prime power such that π(n) > n, then π(n)/4 is an integer that is not greater than n. The multiplicative property of Pisano periods imply thus that
- π(n) ≤ 6n,
with equality if and only if n = 2 · 5r, for r ≥ 1.[5] The first examples are π(10) = 60 and π(50) = 300. If n is not of the form 2 · 5r, then π(n) ≤ 4n.
Tables
The first twelve Pisano periods (sequence A001175 in the OEIS) and their cycles (with spaces before the zeros for readability) are:[6] (using X and E for ten and eleven, respectively)
n | π(n) | number of zeros in the cycle ( | cycle ( | OEIS sequence for the cycle |
---|---|---|---|---|
1 | 1 | 1 | 0 | A000004 |
2 | 3 | 1 | 011 | A011655 |
3 | 8 | 2 | 0112 0221 | A082115 |
4 | 6 | 1 | 011231 | A079343 |
5 | 20 | 4 | 01123 03314 04432 02241 | A082116 |
6 | 24 | 2 | 011235213415 055431453251 | A082117 |
7 | 16 | 2 | 01123516 06654261 | A105870 |
8 | 12 | 2 | 011235 055271 | A079344 |
9 | 24 | 2 | 011235843718 088764156281 | A007887 |
10 | 60 | 4 | 011235831459437 077415617853819 099875279651673 033695493257291 | A003893 |
11 | 10 | 1 | 01123582X1 | A105955 |
12 | 24 | 2 | 011235819X75 055X314592E1 | A089911 |
The first 144 Pisano periods are shown in the following table:
π(n) | +1 | +2 | +3 | +4 | +5 | +6 | +7 | +8 | +9 | +10 | +11 | +12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
0+ | 1 | 3 | 8 | 6 | 20 | 24 | 16 | 12 | 24 | 60 | 10 | 24 |
12+ | 28 | 48 | 40 | 24 | 36 | 24 | 18 | 60 | 16 | 30 | 48 | 24 |
24+ | 100 | 84 | 72 | 48 | 14 | 120 | 30 | 48 | 40 | 36 | 80 | 24 |
36+ | 76 | 18 | 56 | 60 | 40 | 48 | 88 | 30 | 120 | 48 | 32 | 24 |
48+ | 112 | 300 | 72 | 84 | 108 | 72 | 20 | 48 | 72 | 42 | 58 | 120 |
60+ | 60 | 30 | 48 | 96 | 140 | 120 | 136 | 36 | 48 | 240 | 70 | 24 |
72+ | 148 | 228 | 200 | 18 | 80 | 168 | 78 | 120 | 216 | 120 | 168 | 48 |
84+ | 180 | 264 | 56 | 60 | 44 | 120 | 112 | 48 | 120 | 96 | 180 | 48 |
96+ | 196 | 336 | 120 | 300 | 50 | 72 | 208 | 84 | 80 | 108 | 72 | 72 |
108+ | 108 | 60 | 152 | 48 | 76 | 72 | 240 | 42 | 168 | 174 | 144 | 120 |
120+ | 110 | 60 | 40 | 30 | 500 | 48 | 256 | 192 | 88 | 420 | 130 | 120 |
132+ | 144 | 408 | 360 | 36 | 276 | 48 | 46 | 240 | 32 | 210 | 140 | 24 |
Pisano periods of Fibonacci numbers
If n = F (2k) (k ≥ 2), then π(n) = 4k; if n = F (2k + 1) (k ≥ 2), then π(n) = 8k + 4. That is, if the modulo base is a Fibonacci number (≥3) with an even index, the period is twice the index and the cycle has 2 zeros. If the base is a Fibonacci number (≥5) with an odd index, the period is 4 times the index and the cycle has 4 zeros.
k | F (k) | π(F (k)) | first half of cycle (for even k ≥ 4) or first quarter of cycle (for odd k ≥ 4) or all cycle (for k ≤ 3) (with selected second halves or second quarters) |
---|---|---|---|
1 | 1 | 1 | 0 |
2 | 1 | 1 | 0 |
3 | 2 | 3 | 0, 1, 1 |
4 | 3 | 8 | 0, 1, 1, 2, (0, 2, 2, 1) |
5 | 5 | 20 | 0, 1, 1, 2, 3, (0, 3, 3, 1, 4) |
6 | 8 | 12 | 0, 1, 1, 2, 3, 5, (0, 5, 5, 2, 7, 1) |
7 | 13 | 28 | 0, 1, 1, 2, 3, 5, 8, (0, 8, 8, 3, 11, 1, 12) |
8 | 21 | 16 | 0, 1, 1, 2, 3, 5, 8, 13, (0, 13, 13, 5, 18, 2, 20, 1) |
9 | 34 | 36 | 0, 1, 1, 2, 3, 5, 8, 13, 21, (0, 21, 21, 8, 29, 3, 32, 1, 33) |
10 | 55 | 20 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, (0, 34, 34, 13, 47, 5, 52, 2, 54, 1) |
11 | 89 | 44 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, (0, 55, 55, 21, 76, 8, 84, 3, 87, 1, 88) |
12 | 144 | 24 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, (0, 89, 89, 34, 123, 13, 136, 5, 141, 2, 143, 1) |
13 | 233 | 52 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 |
14 | 377 | 28 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 |
15 | 610 | 60 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 |
16 | 987 | 32 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 |
17 | 1597 | 68 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 |
18 | 2584 | 36 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597 |
19 | 4181 | 76 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584 |
20 | 6765 | 40 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181 |
21 | 10946 | 84 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765 |
22 | 17711 | 44 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946 |
23 | 28657 | 92 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711 |
24 | 46368 | 48 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657 |
Pisano periods of Lucas numbers
If n = L (2k) (k ≥ 1), then π(n) = 8k; if n = L (2k + 1) (k ≥ 1), then π(n) = 4k + 2. That is, if the modulo base is a Lucas number (≥3) with an even index, the period is 4 times the index. If the base is a Lucas number (≥4) with an odd index, the period is twice the index.
k | L (k) | π(L (k)) | first half of cycle (for odd k ≥ 2) or first quarter of cycle (for even k ≥ 2) or all cycle (for k = 1) (with selected second halves or second quarters) |
---|---|---|---|
1 | 1 | 1 | 0 |
2 | 3 | 8 | 0, 1, (1, 2) |
3 | 4 | 6 | 0, 1, 1, (2, 3, 1) |
4 | 7 | 16 | 0, 1, 1, 2, (3, 5, 1, 6) |
5 | 11 | 10 | 0, 1, 1, 2, 3, (5, 8, 2, 10, 1) |
6 | 18 | 24 | 0, 1, 1, 2, 3, 5, (8, 13, 3, 16, 1, 17) |
7 | 29 | 14 | 0, 1, 1, 2, 3, 5, 8, (13, 21, 5, 26, 2, 28, 1) |
8 | 47 | 32 | 0, 1, 1, 2, 3, 5, 8, 13, (21, 34, 8, 42, 3, 45, 1, 46) |
9 | 76 | 18 | 0, 1, 1, 2, 3, 5, 8, 13, 21, (34, 55, 13, 68, 5, 73, 2, 75, 1) |
10 | 123 | 40 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, (55, 89, 21, 110, 8, 118, 3, 121, 1, 122) |
11 | 199 | 22 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, (89, 144, 34, 178, 13, 191, 5, 196, 2, 198, 1) |
12 | 322 | 48 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, (144, 233, 55, 288, 21, 309, 8, 317, 3, 320, 1, 321) |
13 | 521 | 26 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 |
14 | 843 | 56 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 |
15 | 1364 | 30 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 |
16 | 2207 | 64 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 |
17 | 3571 | 34 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 |
18 | 5778 | 72 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597 |
19 | 9349 | 38 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584 |
20 | 15127 | 80 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181 |
21 | 24476 | 42 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765 |
22 | 39603 | 88 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946 |
23 | 64079 | 46 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711 |
24 | 103682 | 96 | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657 |
For even k, the cycle has 2 zeros. For odd k, the cycle has only 1 zero, and the second half of the cycle, which is of course equal to the part on the left of 0, consists of alternatingly numbers F(2m + 1) and n − F(2m), with m decreasing.
Number of zeros in the cycle
The number of occurrences of 0 per cycle is 1, 2, or 4. Let p be the number after the first 0 after the combination 0, 1. Let the distance between the 0s be q.
- There is one 0 in a cycle, obviously, if p = 1. This is only possible if q is even or n is 1 or 2.
- Otherwise there are two 0s in a cycle if p2 ≡ 1. This is only possible if q is even.
- Otherwise there are four 0s in a cycle. This is the case if q is odd and n is not 1 or 2.
For generalized Fibonacci sequences (satisfying the same recurrence relation, but with other initial values, e.g. the Lucas numbers) the number of occurrences of 0 per cycle is 0, 1, 2, or 4.
The ratio of the Pisano period of n and the number of zeros modulo n in the cycle gives the rank of apparition or Fibonacci entry point of n. That is, smallest index k such that n divides F(k). They are:
- 1, 3, 4, 6, 5, 12, 8, 6, 12, 15, 10, 12, 7, 24, 20, 12, 9, 12, 18, 30, 8, 30, 24, 12, 25, 21, 36, 24, 14, 60, 30, 24, 20, 9, 40, 12, 19, 18, 28, 30, 20, 24, 44, 30, 60, 24, 16, 12, ... (sequence A001177 in the OEIS)
In Renault's paper the number of zeros is called the "order" of F mod m, denoted , and the "rank of apparition" is called the "rank" and denoted .[7]
According to Wall's conjecture, . If has prime factorization then [7]
Generalizations
The Pisano periods of Pell numbers (or 2-Fibonacci numbers) are
- 1, 2, 8, 4, 12, 8, 6, 8, 24, 12, 24, 8, 28, 6, 24, 16, 16, 24, 40, 12, 24, 24, 22, 8, 60, 28, 72, 12, 20, 24, 30, 32, 24, 16, 12, 24, 76, 40, 56, 24, 10, 24, 88, 24, 24, 22, 46, 16, ... (sequence A175181 in the OEIS)
The Pisano periods of 3-Fibonacci numbers are
- 1, 3, 2, 6, 12, 6, 16, 12, 6, 12, 8, 6, 52, 48, 12, 24, 16, 6, 40, 12, 16, 24, 22, 12, 60, 156, 18, 48, 28, 12, 64, 48, 8, 48, 48, 6, 76, 120, 52, 12, 28, 48, 42, 24, 12, 66, 96, 24, ... (sequence A175182 in the OEIS)
The Pisano periods of Jacobsthal numbers (or (1,2)-Fibonacci numbers) are
- 1, 1, 6, 2, 4, 6, 6, 2, 18, 4, 10, 6, 12, 6, 12, 2, 8, 18, 18, 4, 6, 10, 22, 6, 20, 12, 54, 6, 28, 12, 10, 2, 30, 8, 12, 18, 36, 18, 12, 4, 20, 6, 14, 10, 36, 22, 46, 6, ... (sequence A175286 in the OEIS)
The Pisano periods of (1,3)-Fibonacci numbers are
- 1, 3, 1, 6, 24, 3, 24, 6, 3, 24, 120, 6, 156, 24, 24, 12, 16, 3, 90, 24, 24, 120, 22, 6, 120, 156, 9, 24, 28, 24, 240, 24, 120, 48, 24, 6, 171, 90, 156, 24, 336, 24, 42, 120, 24, 66, 736, 12, ... (sequence A175291 in the OEIS)
The Pisano periods of Tribonacci numbers (or 3-step Fibonacci numbers) are
- 1, 4, 13, 8, 31, 52, 48, 16, 39, 124, 110, 104, 168, 48, 403, 32, 96, 156, 360, 248, 624, 220, 553, 208, 155, 168, 117, 48, 140, 1612, 331, 64, 1430, 96, 1488, 312, 469, 360, 2184, 496, 560, 624, 308, 440, 1209, 2212, 46, 416, ... (sequence A046738 in the OEIS)
The Pisano periods of Tetranacci numbers (or 4-step Fibonacci numbers) are
- 1, 5, 26, 10, 312, 130, 342, 20, 78, 1560, 120, 130, 84, 1710, 312, 40, 4912, 390, 6858, 1560, 4446, 120, 12166, 260, 1560, 420, 234, 1710, 280, 1560, 61568, 80, 1560, 24560, 17784, 390, 1368, 34290, 1092, 1560, 240, 22230, 162800, 120, 312, 60830, 103822, 520, ... (sequence A106295 in the OEIS)
See also generalizations of Fibonacci numbers.
Number theory
Pisano periods can be analyzed using algebraic number theory.
Let be the n-th Pisano period of the k-Fibonacci sequence Fk(n) ( k can be any natural number, these sequences are defined as Fk(0) = 0, Fk(1) = 1, and for any natural number n > 1, Fk(n) = kFk(n-1) + Fk(n-2)). If m and n are coprime, then by the Chinese remainder theorem: two numbers are congruent modulo mn if and only if they are congruent modulo m and modulo n, assuming these latter are coprime. For example, and so Thus it suffices to compute Pisano periods for prime powers (Usually, , unless p is k-Wall-Sun-Sun prime, or k-Fibonacci-Wieferich prime, that is, p2 divides Fk(p-1) or Fk(p+1), where Fk is the k-Fibonacci sequence, for example, 241 is a 3-Wall-Sun-Sun prime, since 2412 divides F3(242).)
For prime numbers p, these can be analyzed by using Binet's formula:
- where is the kth metallic mean
If k2+4 is a quadratic residue modulo p (and , p does not divide k2+4), then and can be expressed as integers modulo p, and thus Binet's formula can be expressed over integers modulo p, and thus the Pisano period divides the totient , since any power (such as ) has period dividing as this is the order of the group of units modulo p.
For k = 1, this first occurs for p = 11, where 42 = 16 ≡ 5 (mod 11) and 2 · 6 = 12 ≡ 1 (mod 11) and 4 · 3 = 12 ≡ 1 (mod 11) so 4 = √5, 6 = 1/2 and 1/√5 = 3, yielding φ = (1 + 4) · 6 = 30 ≡ 8 (mod 11) and the congruence
Another example, which shows that the period can properly divide p − 1, is π1(29) = 14.
If k2+4 is not a quadratic residue modulo p, then Binet's formula is instead defined over the quadratic extension field (Z/p)[√k^2+4], which has p2 elements and whose group of units thus has order p2 − 1, and thus the Pisano period divides p2 − 1. For example, for p = 3 one has π1(3) = 8 which equals 32 − 1 = 8; for p = 7, one has π1(7) = 16, which properly divides 72 − 1 = 48.
This analysis fails for p = 2 and p is a divisor of the squarefree part of k2+4, since in these cases are zero divisors, so one must be careful in interpreting 1/2 or √k^2+4. For p = 2, k2+4 is congruent to 1 mod 2 (for k odd), but the Pisano period is not p − 1 = 1, but rather 3 (in fact, this is also 3 for even k). For p divides the squarefree part of k2+4, the Pisano period is πk(k2+4) = p2-p = p(p − 1), which does not divide p − 1 or p2 − 1.
Fibonacci integer sequences modulo n
One can consider Fibonacci integer sequences and take them modulo n, or put differently, consider Fibonacci sequences in the ring Z/nZ. The period is a divisor of π(n). The number of occurrences of 0 per cycle is 0, 1, 2, or 4. If n is not a prime the cycles include those that are multiples of the cycles for the divisors. For example, for n = 10 the extra cycles include those for n = 2 multiplied by 5, and for n = 5 multiplied by 2.
Table of the extra cycles: (the original Fibonacci cycles are excluded) (using X and E for ten and eleven, respectively)
n | multiples | other cycles | number of cycles (including the original Fibonacci cycles) |
---|---|---|---|
1 | 1 | ||
2 | 0 | 2 | |
3 | 0 | 2 | |
4 | 0, 022 | 033213 | 4 |
5 | 0 | 1342 | 3 |
6 | 0, 0224 0442, 033 | 4 | |
7 | 0 | 02246325 05531452, 03362134 04415643 | 4 |
8 | 0, 022462, 044, 066426 | 033617 077653, 134732574372, 145167541563 | 8 |
9 | 0, 0336 0663 | 022461786527 077538213472, 044832573145 055167426854 | 5 |
10 | 0, 02246 06628 08864 04482, 055, 2684 | 134718976392 | 6 |
11 | 0 | 02246X5492, 0336942683, 044819X874, 055X437X65, 0661784156, 0773X21347, 0885279538, 0997516729, 0XX986391X, 14593, 18964X3257, 28X76 | 14 |
12 | 0, 02246X42682X 0XX8628X64X2, 033693, 0448 0884, 066, 099639 | 07729E873X1E 0EEX974E3257, 1347E65E437X538E761783E2, 156E5491XE98516718952794 | 10 |
Number of Fibonacci integer cycles mod n are:
Notes
- ↑ Weisstein, Eric W. "Pisano Period". MathWorld.
- ↑ On Arithmetical functions related to the Fibonacci numbers. Acta Arithmetica XVI (1969). Retrieved 22 September 2011.
- ↑ A Theorem on Modular Fibonacci Periodicity. Theorem of the Day (2015). Retrieved 7 January 2016.
- ↑ Jiří, Klaška,. "Short remark on Fibonacci-Wieferich primes". Acta Mathematica Universitatis Ostraviensis. 15 (1). ISSN 1214-8148.
- ↑ Freyd & Brown (1992)
- ↑ Sloane, N.J.A. (ed.). "Sequence A001175: graph". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Graph of the cycles modulo 1 to 24. Each row of the image represents a different modulo base n, from 1 at the bottom to 24 at the top. The columns represent the Fibonacci numbers mod n, from F(0) mod n at the left to F(59) mod n on the right. In each cell, the brightness indicates the value of the residual, from dark for 0 to near-white for n−1. Blue squares on the left represent the first period; the number of blue squares is the Pisano number.
- 1 2 "The Fibonacci Sequence Modulo M, by Marc Renault". webspace.ship.edu. Retrieved 2018-08-22.
References
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- Engstrom, H. T. (1931), "On sequences defined by linear recurrence relations", Trans. Am. Math. Soc., 33 (1): 210–218, doi:10.1090/S0002-9947-1931-1501585-5, JSTOR 1989467, MR 1501585
- Falcon, S.; Plaza, A. (2009), "k-Fibonacci sequence modulo m", Chaos, Solitons & Fractals, 41 (1): 497–504, Bibcode:2009CSF....41..497F, doi:10.1016/j.chaos.2008.02.014
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External links
- The Fibonacci sequence modulo m
- A research for Fibonacci numbers
- Fibonacci sequence starts with q, r modulo m
- Johnson, Robert C., Fibonacci resources
- Fibonacci Mystery - Numberphile on YouTube, a video with Dr. James Grime and the University of Nottingham