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Nth Fibonacci number using Pell’s equation

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  • Last Updated : 19 Oct, 2022
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Given an integer N, the task is to find the Nth Fibonacci number.
 Examples:

Input: N = 13 
Output: 144

Input: N = 19 
Output: 2584 

 

Approach: The Nth Fibonacci number can be found using the roots of the pell’s equation. Pells equation is generally of the form (x2) – n(y2) = |1|
Here, consider y2 = x, n = 1. Also, taken positive (+1) in the right-hand side. 
Now the equation becomes x2 – x = 1 which is same as x2 – x – 1 = 0
Here, {x = (pi – qi) / (p – q)} is termed as Nth term of the fibonacci series where i = n – 1 and (p, q) are the roots of the pell’s equation. 
 

To find roots of general quadratic equation (a*x2 + b*x + c = 0). 
x1 = [-b + math.sqrt(b2 – 4*a*c)] / 2*a 
x2 = [-b – math.sqrt(b2 – 4*a*c)] / 2*a 
i.e. 
p = (1 + math.sqrt(5)) / 2 
q = (1 – math.sqrt(5)) / 2 
 

Below is the implementation of the above approach: 

C++




// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to return the
// nth fibonacci number
int fib(int n)
{
    // Assign roots of the pell's
    // equation to p and q
    double p = ((1 + sqrt(5)) / 2);
    double q = ((1 - sqrt(5)) / 2);
    int i = n - 1;
    int x = (int) ((pow(p, i) -
                    pow(q, i)) / (p - q));
    return x;
}
 
// Driver code
int main()
{
    int n = 5;
    cout << fib(n);
}
 
// This code is contributed by PrinciRaj1992

Java




// Java implementation of the approach
class GFG
{
     
// Assign roots of the pell's
// equation to p and q
static double p = ((1 + Math.sqrt(5)) / 2);
static double q = ((1 - Math.sqrt(5)) / 2);
 
// Function to return the
// nth fibonacci number
static int fib(int n)
{
    int i = n - 1;
    int x = (int) ((Math.pow(p, i) -
                    Math.pow(q, i)) / (p - q));
    return x;
}
 
// Driver code
public static void main(String[] args)
{
    int n = 5;
    System.out.println(fib(n));
}
}
 
// This code is contributed by 29AjayKumar

Python3




# Python3 implementation of the approach
import math
 
# Assign roots of the pell's
# equation to p and q
p = (1 + math.sqrt(5)) / 2
q = (1 - math.sqrt(5)) / 2
 
# Function to return the
# nth fibonacci number
def fib(n):
    i = n - 1
    x = (p**i - q**i) / (p - q)
    return int(x)
 
# Driver code
n = 5
print(fib(n))

C#




// C# implementation of the approach
using System;
 
class GFG
{
         
// Assign roots of the pell's
// equation to p and q
static double p = ((1 + Math.Sqrt(5)) / 2);
static double q = ((1 - Math.Sqrt(5)) / 2);
 
// Function to return the
// nth fibonacci number
static int fib(int n)
{
    int i = n - 1;
    int x = (int) ((Math.Pow(p, i) -
                    Math.Pow(q, i)) / (p - q));
    return x;
}
 
// Driver code
static public void Main ()
{
    int n = 5;
    Console.Write(fib(n));
}
}
 
// This code is contributed by @ajit..

Javascript




<script>
 
// Javascript implementation of the approach
 
// Function to return the
// nth fibonacci number
function fib(n)
{
    // Assign roots of the pell's
    // equation to p and q
    let p = ((1 + Math.sqrt(5)) / 2);
    let q = ((1 - Math.sqrt(5)) / 2);
    let i = n - 1;
    let x = parseInt((Math.pow(p, i) -
                    Math.pow(q, i)) / (p - q));
    return x;
}
 
// Driver code
    let n = 5;
    document.write(fib(n));
 
</script>

Output: 

3

 

Time complexity: O(logn) 
Auxiliary Space: O(1)


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