Interesting facts about Fibonacci numbers

27 May, 2022
We know Fibonacci number, Fn = Fn-1 + Fn-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++
C
Java
Python3
C#
PHP
Javascript
// C++ program to demonstrate that sequence of last// digits of Fibonacci numbers repeats after 60.#include <bits/stdc++.h>using namespace std; #define max 100 int main(){    long long int arr[max];    arr[0] = 0;    arr[1] = 1;    int i = 0;     // storing Fibonacci numbers    for (i = 2; i < max; i++) {        arr[i] = arr[i - 1] + arr[i - 2];    }     // Traversing through store numbers    for (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        cout << i << endl;        if ((arr[i] % 10 == 0) && (arr[i + 1] % 10 == 1)) {            break;        }    }    cout << "Sequence is repeating after index " << i         << endl;}
// 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;    int i = 0;     // storing Fibonacci numbers    for (i = 2; i < max; i++)        arr[i] = arr[i - 1] + arr[i - 2];     // Traversing through store numbers    for (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);} // The code is contributed by Gautam goel (gautamgoel962)
// Java program to demonstrate that sequence of last// digits of Fibonacci numbers repeats after 60. class GFG{static int max=100;public static void main(String[] args){    long[] arr=new long[max];    arr[0] = 0;    arr[1] = 1;    int i=0;     // storing Fibonacci numbers    for (i = 2; i < max; i++)        arr[i] = arr[i-1] + arr[i-2];     // Traversing through store numbers    for (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;    }    System.out.println("Sequence is repeating after index "+i);}}// This code is contributed by mits
# Python3 program to demonstrate that sequence of last# digits of Fibonacci numbers repeats after 60.  if __name__=='__main__':    max = 100    arr = [0 for i in range(max)]    arr[0] = 0    arr[1] = 1 # storing Fibonacci numbers    for i in range(2, max):        arr[i] = arr[i - 1] + arr[i - 2]     # Traversing through store numbers    for i in range(1, max - 1):              # 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) and (arr[i + 1] % 10 == 1)):            break     print("Sequence is repeating after index", i) # This code is contributed by# Sanjit_Prasad
// C# program to demonstrate that sequence of last// digits of Fibonacci numbers repeats after 60. class GFG{static int max=100;public static void Main(){    long[] arr=new long[max];    arr[0] = 0;    arr[1] = 1;    int i=0;     // storing Fibonacci numbers    for (i = 2; i < max; i++)        arr[i] = arr[i-1] + arr[i-2];     // Traversing through store numbers    for (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;    }    System.Console.WriteLine("Sequence is repeating after index "+i);}}// This code is contributed by mits
<?php// php program to demonstrate that// sequence of last digits of// Fibonacci numbers repeats after// 60. global $MAX=100     $arr[0] = 0;    $arr[1] = 1;     // storing Fibonacci numbers    for ($i = 2; $i < 100; $i++)        $arr[$i] = $arr[$i-1] +                       $arr[$i-2];     // Traversing through store    // numbers    for ($i = 1; $i <100 - 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;    }    echo "Sequence is repeating after",                         " index ", $i; // This code is contributed by ajit?>
<script> // Javascript program to demonstrate that// sequence of last digits of Fibonacci// numbers repeats after 60.    var max = 100; var arr = Array(max).fill(0);arr[0] = 0;arr[1] = 1;var i = 0; // Storing Fibonacci numbersfor(i = 2; i < max; i++)    arr[i] = arr[i - 1] + arr[i - 2]; // Traversing through store numbersfor(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;} // Driver codedocument.write("Sequence is repeating after index " + i); // This code is contributed by gauravrajput1 </script>
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. 
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C
Java
Python3
C#
PHP
Javascript
// C++ program to demonstrate divisibility of Fibonacci// numbers.#include<iostream>using namespace std;#define MAX 90 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    cout<<"Index of Fibonacci numbers divisible by"           " 2 are :\n";    for (int i = 0; i < c1; i++)        cout<<" "<< index1[i];    cout<<"\n";     cout<<"Index of Fibonacci number divisible by"           " 3 are :\n";    for (int i = 0; i < c2; i++)        cout<<"  "<< index2[i];    cout<<"\n";     cout<<"Index of Fibonacci number divisible by"           " 5 are :\n";    for (int i = 0; i < c3; i++)        cout<<"  "<< index3[i];    cout<<"\n";     cout<<"Index of Fibonacci number divisible by"           " 8 are :\n";    for (int i = 0; i < c4; i++)        cout<<" "<<index4[i];    cout<<"\n";} // This code is contributed by shivanisinghss2110
// C program to demonstrate divisibility of Fibonacci// numbers.#include<stdio.h>#define MAX 90 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");}
// Java program to demonstrate divisibility of Fibonacci// numbers. class GFG{static int MAX=90; // Driver codepublic static void main(String[] args){    // indexes variable stores index of number that    // is divisible by 2, 3, 5 and 8    long[] arr=new long[MAX];    long[] index1=new long[MAX];    long[] index2=new long[MAX];    long[] index3=new long[MAX];    long[] index4=new long[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    System.out.print("Index of Fibonacci numbers divisible by" +        " 2 are :\n");    for (int i = 0; i < c1; i++)        System.out.print(index1[i] + " ");    System.out.print("\n");     System.out.print("Index of Fibonacci number divisible by" +        " 3 are :\n");    for (int i = 0; i < c2; i++)        System.out.print(index2[i] + " ");    System.out.print("\n");     System.out.print("Index of Fibonacci number divisible by" +        " 5 are :\n");    for (int i = 0; i < c3; i++)        System.out.print(index3[i] + " ");    System.out.print("\n");     System.out.print("Index of Fibonacci number divisible by" +        " 8 are :\n");    for (int i = 0; i < c4; i++)        System.out.print(index4[i] + " ");    System.out.print("\n");}} // This code is contributed by mits
# Python3 program to demonstrate divisibility# of Fibonacci numbers.MAX = 90; # indexes variable stores index of number# that is divisible by 2, 3, 5 and 8arr = [0] * (MAX);index1 = [0] * (MAX);index2 = [0] * (MAX);index3 = [0] * (MAX);index4 = [0] * (MAX); # storing fibonacci numbersarr[0] = 0;arr[1] = 1;for i in range(2, MAX):    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 8c1, c2, c3, c4 = 0, 0, 0, 0; # separating fibonacci number into# their respective arrayfor i in range(MAX):    if (arr[i] % 2 == 0):        index1[c1] = i;        c1 += 1;    if (arr[i] % 3 == 0):        index2[c2] = i;        c2 += 1;    if (arr[i] % 5 == 0):        index3[c3] = i;        c3 += 1;    if (arr[i] % 8 == 0):        index4[c4] = i;        c4 += 1; # printing index arraysprint("Index of Fibonacci numbers",           "divisible by 2 are :");for i in range(c1):    print(index1[i], end = " ");print(""); print("Index of Fibonacci number",          "divisible by 3 are :");for i in range(c2):    print(index2[i], end = " ");print(""); print("Index of Fibonacci number",          "divisible by 5 are :");for i in range(c3):    print(index3[i], end = " ");print(""); print("Index of Fibonacci number",          "divisible by 8 are :");for i in range(c4):    print(index4[i], end = " ");print(""); # This code is contributed by mits
// C# program to demonstrate divisibility// of Fibonacci numbers. class GFG{static int MAX = 90; static void Main(){    // indexes variable stores index of number that    // is divisible by 2, 3, 5 and 8    long[] arr = new long[MAX];    long[] index1 = new long[MAX];    long[] index2 = new long[MAX];    long[] index3 = new long[MAX];    long[] index4 = new long[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    System.Console.Write("Index of Fibonacci numbers" +                    "divisible by 2 are :\n");    for (int i = 0; i < c1; i++)        System.Console.Write(index1[i]+" ");    System.Console.Write("\n");     System.Console.Write("Index of Fibonacci number "+                        " divisible by 3 are :\n");    for (int i = 0; i < c2; i++)        System.Console.Write(index2[i]+" ");    System.Console.Write("\n");     System.Console.Write("Index of Fibonacci number "+        "divisible by 5 are :\n");    for (int i = 0; i < c3; i++)        System.Console.Write(index3[i]+" ");    System.Console.Write("\n");     System.Console.Write("Index of Fibonacci number "+        "divisible by 8 are :\n");    for (int i = 0; i < c4; i++)        System.Console.Write(index4[i]+" ");    System.Console.Write("\n");}} // This code is contributed by mits
<?php// PHP program to demonstrate divisibility// of Fibonacci numbers.$MAX = 90; // indexes variable stores index of number// that is divisible by 2, 3, 5 and 8$arr = array($MAX);$index1 = array($MAX);$index2 = array($MAX);$index3 = array($MAX);$index4 = array($MAX); // storing fibonacci numbers$arr[0] = 0;$arr[1] = 1;for ($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$c1 = 0;$c2 = 0;$c3 = 0;$c4 = 0; // separating fibonacci number into// their respective arrayfor ($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++] = $i;} // printing index arraysecho "Index of Fibonacci numbers divisible by" .                                " 2 are :\n";for ($i = 0; $i < $c1; $i++)    echo $index1[$i] . " ";echo "\n"; echo "Index of Fibonacci number divisible by" .                                " 3 are :\n";for ($i = 0; $i < $c2; $i++)    echo $index2[$i] . " ";echo "\n"; echo "Index of Fibonacci number divisible by" .    " 5 are :\n";for ($i = 0; $i < $c3; $i++)    echo $index3[$i] . " ";echo "\n"; echo "Index of Fibonacci number divisible by" .    " 8 are :\n";for ($i = 0; $i < $c4; $i++)    echo $index4[$i] . " ";echo "\n"; // This code is contributed by mits?>
// JavaScript program to demonstrate divisibility of Fibonacci// numbers. var MAX=90; // Driver code     // indexes variable stores index of number that    // is divisible by 2, 3, 5 and 8    var arr=new Array(MAX);    var index1= new Array(MAX);    var index2= new Array(MAX);    var index3= new Array(MAX);    var index4= new Array(MAX);     // storing fibonacci numbers    arr[0] = 0;    arr[1] = 1;    for (var 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    var c1 = 0, c2 = 0, c3 = 0, c4 = 0;     // separating fibonacci number into their    // respective array    for (var 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    document.write("Index of Fibonacci numbers divisible by" +        " 2 are :\n");    for (var i = 0; i < c1; i++)        document.write(index1[i] + " ");    document.write("\n");     document.write("Index of Fibonacci number divisible by" +        " 3 are :\n");    for (var i = 0; i < c2; i++)        document.write(index2[i] + " ");    document.write("\n");     document.write("Index of Fibonacci number divisible by" +        " 5 are :\n");    for (var i = 0; i < c3; i++)        document.write(index3[i] + " ");    document.write("\n");     document.write("Index of Fibonacci number divisible by" +        " 8 are :\n");    for (var i = 0; i < c4; i++)        document.write(index4[i] + " ");    document.write("\n");   // This code is contributed by shivanisinghss2110
Output: 
Index of Fibonacci numbers 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 
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 
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 
Index of Fibonacci number divisible by 8 are :
0 6 12 18 24 30 36 42 48
54 60 66 72 78 84 
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++
C
Java
Python3
C#
Javascript
// 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<iostream>using namespace std;#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];     cout<<"Fibonacci numbers divisible by "          "their indexes are :\n";    for (int i = 1; i < MAX; i++)        if (arr[i] % i == 0)            cout<<"  "<< i;} // This code is contributed by shivanisinghss2110
// 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);}
// Java program to demonstrate that Fibonacci numbers// that are divisible by their indexes have indexes// as either power of 5 or multiple of 12. class GFG{ static int MAX = 100; public static void main(String[] args){    // storing Fibonacci numbers    long[] arr = new long[MAX];    arr[0] = 0;    arr[1] = 1;    for (int i = 2; i < MAX; i++)        arr[i] = arr[i - 1] + arr[i - 2];     System.out.print("Fibonacci numbers divisible by "+        "their indexes are :\n");    for (int i = 1; i < MAX; i++)        if (arr[i] % i == 0)            System.out.print(i + " ");}} // This code is contributed by mits
# Python3 program to demonstrate that Fibonacci numbers# that are divisible by their indexes have indexes# as either power of 5 or multiple of 12. if __name__=='__main__':    MAX = 100# storing Fibonacci numbers    arr = [0 for i in range(MAX)]    arr[0] = 0    arr[1] = 1    for i in range(2, MAX):        arr[i] = arr[i - 1] + arr[i - 2]     print("Fibonacci numbers divisible by their indexes are :")    for i in range(1, MAX):        if(arr[i] % i == 0):            print(i,end=" ") # This code is contributed by# Sanjit_Prasad
// C# program to demonstrate that Fibonacci// numbers that are divisible by their// indexes have indexes as either power of 5// or multiple of 12.using System; class GFG{static int MAX = 100;static void Main(){    // storing Fibonacci numbers    long[] arr = new long[MAX];    arr[0] = 0;    arr[1] = 1;    for (int i = 2; i < MAX; i++)        arr[i] = arr[i - 1] + arr[i - 2];     Console.Write("Fibonacci numbers divisible by " +                           "their indexes are :\n");    for (int i = 1; i < MAX; i++)        if (arr[i] % i == 0)            System.Console.Write(i+" ");}} // This code is contributed by mits
// JavaScript program to demonstrate that Fibonacci numbers// that are divisible by their indexes have indexes// as either power of 5 or multiple of 12. var MAX = 100;      // storing Fibonacci numbers    var arr = new Array(MAX);    arr[0] = 0;    arr[1] = 1;    for (var i = 2; i < MAX; i++)        arr[i] = arr[i - 1] + arr[i - 2];     document.write("Fibonacci numbers divisible by their indexes are :");    for (var i = 1; i < MAX; i++)        if (arr[i] % i == 0)            document.write(i + " ");  // This code is contributed by shivanisinghss2110
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.htmlThis article is contributed by Aditya Kumar. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. 
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