Test if a Number belongs...

To determine if a given number belongs to the Fibonacci series, there are several methods you can use. Here are the most common approaches:

1. Mathematical Formula

One efficient way to check if a number $$ N $$ is a Fibonacci number is by using a mathematical property. A number $$ N $$ is a Fibonacci number if and only if one or both of $$ 5N^2 + 4 $$ or $$ 5N^2 - 4 $$ is a perfect square. This method is based on the properties of Fibonacci numbers and can be implemented as follows:

Pseudocode

def is_perfect_square(x):
    s = int(math.sqrt(x))
    return s * s == x

def is_fibonacci(n):
    return is_perfect_square(5 * n * n + 4) or is_perfect_square(5 * n * n - 4)

Explanation

1. Calculate $$ 5N^2 + 4 $$ and $$ 5N^2 - 4 $$.
2. Check if either of these values is a perfect square.
3. If either is a perfect square, $$ N $$ is a Fibonacci number.

This method is efficient and has a constant time complexity $$ O(1) $$ for checking if a number is a Fibonacci number[1][2][10].

2. Iterative Approach

Another method is to generate Fibonacci numbers iteratively until the generated number is greater than or equal to $$ N $$. If $$ N $$ is found during this process, it is a Fibonacci number.

Pseudocode

def is_fibonacci(n):
    if n == 0 or n == 1:
        return True
    a, b = 0, 1
    while b < n:
        a, b = b, a + b
    return b == n

Explanation

1. Initialize the first two Fibonacci numbers $$ a = 0 $$ and $$ b = 1 $$.
2. Generate the next Fibonacci number by summing the previous two.
3. Continue this process until the generated number is greater than or equal to $$ N $$.
4. If the generated number equals $$ N $$, then $$ N $$ is a Fibonacci number.

This method has a time complexity of $$ O(\log N) $$ due to the exponential growth of Fibonacci numbers[3][8][10].

3. Using Binet's Formula

Binet's formula provides a direct way to compute the $$ n $$-th Fibonacci number using the golden ratio $$ \phi $$. However, it is more commonly used to generate Fibonacci numbers rather than to check if a number is a Fibonacci number.

Pseudocode

import math

def fib_binet(n):
    phi = (1 + math.sqrt(5)) / 2
    psi = (1 - math.sqrt(5)) / 2
    return int((phi**n - psi**n) / math.sqrt(5))

Explanation

1. Calculate the $$ n $$-th Fibonacci number using the golden ratio $$ \phi $$ and its conjugate...