Sum of Square Numbers
Problem Statement
Given a non-negative integer c, the task is to determine if there exist two integers, a and b, such that the sum of the squares of a and b equals c. This means one needs to confirm whether the equation a^2 + b^2 = c holds true for any pair of integers (a, b).
Examples
Example 1
Input:
EXAMPLE_ENTITY
Output:
EXAMPLE_ENTITY
Explanation:
EXAMPLE_ENTITY
Example 2
Input:
EXAMPLE_ENTITY
Output:
EXAMPLE_ENTITY
Constraints
0 <= c <= 231 - 1
Approach and Intuition
The problem is essentially about finding if a given integer can be expressed as the sum of squares of two integers. The brute-force way to approach this would be to check all pairs (a, b) such that a^2 + b^2 could potentially equal c. However, we can optimize this process by leveraging some math and constraints:
Since we know that squaring a number cannot be negative, both
aandbshould range from0tosqrt(c). This is because for anyaorbgreater thansqrt(c),a^2orb^2alone would be greater thanc.Start with
a = 0and computeb^2 = c - a^2. Ifb^2turns out to be a perfect square andbis an integer, then we have found such a pair, and we can return true. If not, incrementaand check again.Continue this process until
a > sqrt(c). If by the end no suitable pair(a, b)is found, return false.
This method ensures that we are not unnecessarily checking pairs that would obviously not satisfy the condition, hence optimizing our search space and time complexity.
Solutions
CPP
The code provided addresses the problem of determining whether a given integer can be expressed as a sum of two square numbers. This solution is implemented in C++ within a class named Solution.
The method canBeSumOfSquares takes an integer value as input and uses a for loop to iterate through possible base numbers from 2 up to the square root of the value. Within the loop, a variable factorCount is initialized to zero to count the factors of value that match the current base.
During each iteration, while the remainder of the division of value by base is zero, factorCount is incremented, and value is divided by base. This while loop continues until value can no longer be divided by base without a remainder.
After exiting the while loop, the code checks if base, when modulo 4, equals 3 and factorCount is odd. If this condition is true, the function immediately returns false, indicating that value cannot be expressed as a sum of two squares under these conditions.
Finally, after the loop ends, the method returns true if value modulo 4 is not equal to 3, meaning the original input value or its residual factor, when tested, does not prohibit value from being expressed as the sum of two squares. Conversely, it returns false if any residual value does match the condition of being congruent to 3 modulo 4.
This function essentially utilizes mathematical properties specific to the quadratic forms and is effective in determining the sum of squares possibility under given constraints.
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