Cassini’s Identity (original) (raw)

Last Updated : 23 Jul, 2025

Given a number N, the task is to evaluate below expression. Expected time complexity is O(1).

f(n-1)*f(n+1) - f(n)*f(n)

Where f(n) is the n-th Fibonacci number with n >= 1. First few Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, ...........i.e. (considering 0 as 0th Fibonacci number) Examples :

Input : n = 5 Output : -1 f(5-1=4) = 3 f(5+1=6) = 8 f(5)f(5)= 55 = 25 f(4)*f(6)- f(5)*f(5)= 24-25= -1

Although the task is simple i.e. find n-1th, nth and (n+1)-th Fibonacci numbers. Evaluate the expression and display the result. But this can be done in O(1) time using Cassini’s Identity which states that:

       f(n-1)*f(n+1) - f(n*n) = (-1)^n 

So, we don't need to calculate any Fibonacci term,the only thing is to check whether n is even or odd. How does above formula work? The formula is based on matrix representation of Fibonacci numbers.

C/C++

 C++ `

// C++ implementation to demonstrate working // of Cassini’s Identity #include <bits/stdc++.h> using namespace std;

// Returns (-1)^n int cassini(int n) { return (n & 1) != 0 ? -1 : 1; }

// Driver Method int main() { int n = 5; cout << (cassini(n));

return 0;

}

// This code is contributed by phasing17

Java

// Java implementation to demonstrate working // of Cassini’s Identity

class Gfg { // Returns (-1)^n static int cassini(int n) { return (n & 1) != 0 ? -1 : 1; }

// Driver method
public static void main(String args[])
{
     int n = 5;
     System.out.println(cassini(n));
}

}

Python3

Python implementation

to demonstrate working

of Cassini’s Identity

Returns (-1)^n

def cassini(n):

return -1 if (n & 1) else 1

Driver program

n = 5 print(cassini(n))

This code is contributed

by Anant Agarwal.

C#

// C# implementation to demonstrate // working of Cassini’s Identity using System;

class GFG {

// Returns (-1) ^ n
static int cassini(int n)
{
   return (n & 1) != 0 ? -1 : 1;
} 

// Driver Code
public static void Main()
{
     int n = 5;
     Console.Write(cassini(n));
}

}

// This code is contributed by Nitin Mittal.

PHP

JavaScript

`

Output :

-1

Time complexity: O(1) since only constant operations are performed

Auxiliary Space: O(1)

Reference : https://en.wikipedia.org/wiki/Cassini_and_Catalan_identities