Problem Statement
Square Number Decomposition
You are given a non-negative integer c. Your task is to determine whether there exist two integers a and b such that:
a² + b² = c
In other words, you need to decide if the given number can be expressed as the sum of two perfect squares.
Examples
Example 1:
Input: c = 5
Output: true
Explanation: 1² + 2² = 1 + 4 = 5
Example 2:
Input: c = 3
Output: false
Explanation: There are no two integers a and b such that a² + b² = 3
Example 3:
Input: c = 25
Output: true
Explanation: 3² + 4² = 9 + 16 = 25
Constraints
- 0 <= c <= 2³¹ - 1
Problem Breakdown
To solve this problem, we need to:
- The problem can be approached using a two-pointer technique, starting from the smallest and largest possible values
- The maximum value of a or b cannot exceed the square root of c
- The problem is related to the Pythagorean theorem and Fermat's theorem on the sum of two squares
- A number can be expressed as the sum of two squares if and only if its prime factorization contains even powers of primes of the form 4k+3
- Binary search can be used to check if c - a² is a perfect square for each possible value of a
String Problems
Learning Objective
Apply string manipulation concepts to solve a real-world problem.
Square Number Decomposition
You are given a non-negative integer c. Your task is to determine whether there exist two integers a and b such that:
a² + b² = c
In other words, you need to decide if the given number can be expressed as the sum of two perfect squares.
Examples
Example 1:
1² + 2² = 1 + 4 = 5
Example 2:
There are no two integers a and b such that a² + b² = 3
Example 3:
3² + 4² = 9 + 16 = 25
Problem Breakdown
Constraints:
- 0 <= c <= 2³¹ - 1
Key Insights:
The problem can be approached using a two-pointer technique, starting from the smallest and largest possible values
The maximum value of a or b cannot exceed the square root of c
The problem is related to the Pythagorean theorem and Fermat's theorem on the sum of two squares
A number can be expressed as the sum of two squares if and only if its prime factorization contains even powers of primes of the form 4k+3
Binary search can be used to check if c - a² is a perfect square for each possible value of a
Real-World Application
This problem has several practical applications:
Number Theory
Understanding the properties of numbers and their representations in mathematics.
Pythagorean Triples
Finding integer solutions to the Pythagorean theorem, which has applications in geometry and physics.
Cryptography
Some cryptographic algorithms rely on properties of numbers, including their representation as sums of squares.
Computer Graphics
Distance calculations in 2D space often involve sums of squares, which is useful in rendering and collision detection.
Optimization Problems
Many optimization algorithms involve minimizing the sum of squared errors or distances.