(Square numbers) Find the first 5 square numbers that are greater than Long.MAX_VALUE. A square number is a number in the form of n2 . For example, 4, 9, and 16 are square numbers. Find an efficient approach to run your program fast.

Since Long.MAX_VALUE is the largest possible value for a long integer in Java, we cannot calculate the square of any integer greater than or equal to its square root without exceeding Long.MAX_VALUE. Therefore, we need to find the first 5 square numbers that are greater than Long.MAX_VALUE using a different approach.

One efficient approach is to use the BigInteger class in Java, which allows us to perform arithmetic operations with arbitrarily large integers. We can create a BigInteger variable initialized with the square root of Long.MAX_VALUE and then increment it until we find the first 5 square numbers that are greater than Long.MAX_VALUE.

Here's the Java code to implement this approach:

import java.math.BigInteger;

public class SquareNumbers {
    public static void main(String[] args) {
        BigInteger maxLong = BigInteger.valueOf(Long.MAX_VALUE);
        BigInteger n = sqrt(maxLong).add(BigInteger.ONE); // start with sqrt(Long.MAX_VALUE) + 1

        int count = 0;
        while (count < 5) {
            BigInteger square = n.multiply(n);
            if (square.compareTo(maxLong) > 0) { // if square > Long.MAX_VALUE
                System.out.println(square);
                count++;
            }
            n = n.add(BigInteger.ONE);
        }
    }

    // helper method to calculate the square root of a BigInteger
    private static BigInteger sqrt(BigInteger n) {
        BigInteger a = BigInteger.ONE;
        BigInteger b = n.shiftRight(5).add(BigInteger.valueOf(8));
        while (b.compareTo(a) >= 0) {
            BigInteger mid = a.add(b).shiftRight(1);
            if (mid.multiply(mid).compareTo(n) > 0) {
                b = mid.subtract(BigInteger.ONE);
            } else {
                a = mid.add(BigInteger.ONE);
            }
        }
        return a.subtract(BigInteger.ONE);
    }
}

The sqrt() method is a helper method to calculate the square root of a BigInteger using the binary search algorithm. The main() method initializes n with the square root of Long.MAX_VALUE plus one, and then uses a while loop to increment n and check if its square is greater than Long.MAX_VALUE. Once we find 5 such square numbers, we print them out and exit the program.