Saint-Exupery Numbers in Java
The Saint-Exupery Number is the product of the three sides of the Pythagorean triangle.There must exist positive integers a, b, and c such that, for a given number N, a^2 + b^2 = c^2 and a.b.c is equivalent to N.
It can be better understood by looking at an example.
Example 1:
Input: N = 60
Output: Yes, 60 is a Saint-Exupery number.
So, for N = 60, the numbers 3, 4, and 5 form a Pythagorean triplet, and N is a Saint-Exupery number. Other examples, such as N = 480, 780, 1620, etc., can be proven to be Saint-Exupery numbers by identifying corresponding Pythagorean triplets that meet the above conditions. Finding appropriate values for a, b, and c for each given N requires trial and error.
Naïve Approach:
The naive approach for identifying Saint-Exupery numbers involves carefully analyzing each triplet combination to see if it meets the requirements for a Saint-Exupery number. This generally involves iterating through various possibilities using multiple nested loops.
Let's see the implementation of the Naive approach :
FileName: SaintExuperyNumbers.java
public class SaintExuperyNumbers {
// Function to check if a number is a perfect square
private static boolean isPerfectSquare(int n) {
int sqrt = (int) Math.sqrt(n);
return sqrt * sqrt == n;
}
// Function to check if a number is a Saint-Exupery Number using the naive approach
private static boolean isSaintExuperyNumberNaive(int n) {
for (int a = 1; a < n; a++) {
for (int b = a + 1; b < n; b++) {
int cSquared = a * a + b * b;
if (isPerfectSquare(cSquared) && a * b * (int)Math.sqrt(cSquared) == n) {
return true;
}
}
}
return false;
}
public static void main(String[] args) {
int[] saintExuperyNumbers = {60, 480, 780, 1620, 2040};
for (int number : saintExuperyNumbers) {
if (isSaintExuperyNumberNaive(number)) {
System.out.println(number + " is a Saint-Exupery Number");
} else {
System.out.println(number + " is not a Saint-Exupery Number");
}
}
}
}
Output:
60 is a Saint-Exupery Number
480 is a Saint-Exupery Number
780 is a Saint-Exupery Number
1620 is a Saint-Exupery Number
2040 is a Saint-Exupery Number
Complexity analysis: In this naive approach ,it has a time complexity of O(n^3) due to the nested loops. Because the naive approach uses a constant amount of additional space regardless of the size of the input, its space complexity is O(1). The number of Saint-Exupery numbers being checked has no impact on it.
Optimized or Efficient approach:
When searching for Saint-Exupery numbers, the most efficient method is to minimize the number of computations or iterations required to determine whether a given number is a Saint-Exupery number. This could include optimizations or mathematical insights that help the algorithm find solutions more quickly and with less computational overhead.
Let's see the implementation of an optimized or Efficient approach :
FileName: SaintExuperyNumbers1.java
public class SaintExuperyNumbers1 {
// Function to check if a number is a Saint-Exupery Number using the efficient approach
private static boolean isSaintExuperyNumberEfficient(int n) {
for (int i = 1; i <= n / 3; i++) {
for (int j = i + 1; j <= n / 2; j++) {
int k = n / i / j;
if (i * i + j * j == k * k) {
return true;
}
}
}
return false;
}
public static void main(String[] args) {
int[] saintExuperyNumbers = {60, 480, 780, 1620, 2040};
for (int number : saintExuperyNumbers) {
if (isSaintExuperyNumberEfficient(number)) {
System.out.println(number + " is a Saint-Exupery Number");
} else {
System.out.println(number + " is not a Saint-Exupery Number");
}
}
}
}
Output:
60 is a Saint-Exupery Number
480 is a Saint-Exupery Number
780 is a Saint-Exupery Number
1620 is a Saint-Exupery Number
2040 is a Saint-Exupery Number
Complexity Analysis: Due to two nested loops in the `isSaintExuperyNumberEfficient` function, where n is the input number, this approach has an O(n^2) time complexity. This method's space complexity is O(1) since the additional space required is fixed and independent of the size of the input. The algorithm stores variables like i, j, and k in a constant amount of space.