N as the Number of Ways to Write 4 as a Sum of Squares

When given a positive integer N, we want to find the number of distinct ways that 4 can be written as the sum of squares of four non-negative integers. In this article, we will explore different approaches to solve this problem and provide implementation examples in Python.

Problem Statement

Given N, we are looking for the number of solutions to the equation:

N = a^2 + b^2 + c^2 + d^2

where a, b, c, and d are non-negative integers.

Approach 1: Brute Force

One simple and straightforward approach is to use brute force to iterate through all possible combinations of a, b, c, and d. We can start with four nested loops, each ranging from 0 to the square root of N. This approach has a time complexity of O(sqrt(N)^4) and can be implemented as follows:

def count_ways(N):
    count = 0
    for a in range(int(N**0.5) + 1):
        for b in range(int(N**0.5) + 1):
            for c in range(int(N**0.5) + 1):
                for d in range(int(N**0.5) + 1):
                    if a**2 + b**2 + c**2 + d**2 == N:
                        count += 1
    return count

Approach 2: Utilizing Number Theory

By utilizing number theory concepts, we can significantly improve the time complexity of finding the number of ways to write 4 as a sum of squares. We can observe that a number can be written as a sum of squares of four non-negative integers if and only if it satisfies certain conditions. These conditions include:

• The number must not have any prime factor of the form (4k + 3) with an odd exponent.
• If the number has a prime factor of the form (4k + 3) with an even exponent, it must not have any prime factors of the form (4k + 1).

We can use this information to develop a more efficient algorithm to count the number of ways. The time complexity of this approach is O(sqrt(N)*log(N)), which is much faster than the brute force method for large values of N.

def count_ways(N):
    count = 0
    
    # Handle factors of 2 separately
    while N % 2 == 0:
        count += 1
        N //= 2
    
    # Check other prime factors
    p = 3
    while p*p <= N:
        if N % p == 0:
            count += 1
            while N % p == 0:
                N //= p
        p += 2
    
    if N > 1:
        count += 1
    
    return count

Conclusion

In this article, we explored different approaches to find the number of ways to write 4 as a sum of squares of four non-negative integers. We presented a brute force approach and an efficient algorithm utilizing number theory concepts. Depending on the value of N, the efficient algorithm can provide a significant improvement in performance compared to the brute force method. It is important to understand the problem requirements and choose the appropriate approach for the given situation.