# Nth Fibonacci number using Pell’s equation

Given an integer N, the task is to find the Nth Fibonacci number.

Input: N = 13
Output: 144

Input: N = 19
Output: 2584

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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 fibonaccci 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 ` `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.. `

Output:

```3
```

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