We know Fibonacci number, **F**n = **F**n-1 + **F**n-2.

First few Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …. .

Here are some interesting facts about Fibonacci number :

**1. Pattern in Last digits of Fibonacci numbers :**

Last digits of first few Fibonacci Numbers are :

0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, ...

The series of last digits repeats with a cycle length of 60 (Refer this for explanations of this result).

// C program to demonstrate that sequence of last // digits of Fibonacci numbers repeats after 60. #include<stdio.h> #define max 100 int main() { long long int arr[max]; arr[0] = 0; arr[1] = 1; // storing Fibonacci numbers for (int i = 2; i < max; i++) arr[i] = arr[i-1] + arr[i-2]; // Traversing through store numbers for (int i = 1; i < max - 1; i++) { // Since first two number are 0 and 1 // so, if any two consecutive number encounter 0 and 1 // at their unit place, then it clearly means that // number is repeating/ since we just have to find // the sum of previous two number if ((arr[i] % 10 == 0) && (arr[i+1] % 10 == 1)) break; } printf("Sequence is repeating after index %d", i); }

Output:

Sequence is repeating after index 60

**2. Factors of Fibonacci number :** On careful observation, we can observe the following thing :

- Every 3-rd Fibonacci number is a multiple of 2
- Every 4-th Fibonacci number is a multiple of 3
- Every 5-th Fibonacci number is a multiple of 5
- Every 6-th Fibonacci number is a multiple of 8

Refer this for details.

// C program to demonstrate divisibility of Fibonacci // numbers. #include<stdio.h> #define MAX 100 int main() { // indexes variable stores index of number that // is divisible by 2, 3, 5 and 8 long long int arr[MAX], index1[MAX], index2[MAX]; long long int index3[MAX], index4[MAX]; // storing fibonacci numbers arr[0] = 0; arr[1] = 1; for (int i = 2; i < MAX; i++) arr[i] = arr[i-1] + arr[i-2]; // c1 keeps track of number of index of number // divisible by 2 and others c2, c3 and c4 for // 3, 5 and 8 int c1 = 0, c2 = 0, c3 = 0, c4 = 0; // separating fibonacci number into their // respective array for (int i = 0; i < MAX; i++) { if (arr[i] % 2 == 0) index1[c1++] = i; if (arr[i] % 3 == 0) index2[c2++] = i; if (arr[i] % 5 == 0) index3[c3++] = i; if (arr[i] % 8 == 0) index4[c4++] = i; } // printing index arrays printf("Index of Fibonacci numbers divisible by" " 2 are :\n"); for (int i = 0; i < c1; i++) printf("%d ", index1[i]); printf("\n"); printf("Index of Fibonacci number divisible by" " 3 are :\n"); for (int i = 0; i < c2; i++) printf("%d ", index2[i]); printf("\n"); printf("Index of Fibonacci number divisible by" " 5 are :\n"); for (int i = 0; i < c3; i++) printf("%d ", index3[i]); printf("\n"); printf("Index of Fibonacci number divisible by" " 8 are :\n"); for (int i = 0; i < c4; i++) printf("%d ", index4[i]); printf("\n"); }

Output:

Index of Fibonacci number divisible by 2 are : 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99 Index of Fibonacci number divisible by 3 are : 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 98 Index of Fibonacci number divisible by 5 are : 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 Index of Fibonacci number divisible by 8 are : 0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96

**3. Fibonacci number with index number factor :** We have some Fibonacci number like F(1) = 1 which is divisible by 1, F(5) = 5 which is divisible by 5, F(12) = 144 which is divisible by 12, F(24) = 46368 which is divisible by 24, F(25) = 75025 which is divisible by 25. This type of index number follow a certain pattern. First, let’s keep a look on those index number :

1, 5, 12, 24, 25, 36, 48, 60, 72, 84, 96, 108, 120, 125, 132, …..

On observing it, this series is made up of every number that is multiple of 12 as well as all the number that satisfies the condition of pow(5, k), where k = 0, 1, 2, 3, 4, 5, 6, 7, …….

// C program to demonstrate that Fibonacci numbers // that are divisible by their indexes have indexes // as either power of 5 or multiple of 12. #include<stdio.h> #define MAX 100 int main() { // storing Fibonacci numbers long long int arr[MAX]; arr[0] = 0; arr[1] = 1; for (int i = 2; i < MAX; i++) arr[i] = arr[i-1] + arr[i-2]; printf("Fibonacci numbers divisible by " "their indexes are :\n"); for (int i = 1; i < MAX; i++) if (arr[i] % i == 0) printf("%d ", i); }

Output:

Fibonacci numbers divisible by their indexes are : 1 5 12 24 25 36 48 60 72 96

**4. Value of f(n-1)*f(n+1) – f(n)*f(n) is (-1) ^{n}**. Please refer Cassini’s Identity for details.

**Reference :**

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibmaths.html

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