Step by Step solution about How to resolve the algorithm Hickerson series of almost integers step by step in the Java programming language

Purpose: The provided Java code determines whether a given integer is "almost integer" based on a mathematical property related to the binary logarithm.

Explanation:

1. Importing Necessary Libraries:

   • The code imports the BigDecimal and BigInteger classes from the java.math package for high-precision decimal and integer operations.

2. The almostInteger() Method:

   • The method takes an integer n as an input and checks if it satisfies a certain mathematical condition.
   • It calculates two decimal values, a and b, and then examines the fractional part of the resulting value.

3. Calculating a and b:

   • a is calculated by raising 2 to the power of (n + 1) and multiplying it by 2. This effectively represents 2^(n+1).
   • b is calculated by dividing the factorial of n (i.e., the product of all positive integers less than or equal to n) by a.

4. Checking the Fractional Part:

   • The fractional part of b is extracted by moving the decimal point one place to the right and converting it to a BigInteger.
   • The resulting BigInteger is then moduloed by 10 to obtain the last digit of the fractional part.
   • If this last digit is either 0 or 9, the method returns true. Otherwise, it returns false.

5. The LN2 Constant:

   • The code defines a constant LN2 with an approximation of the natural logarithm of 2. This constant is used to compute a accurately.

6. Main Execution:

   • In the main() method, the code iterates through integers from 1 to 17 and calls the almostInteger() method for each integer.
   • It prints the results, indicating whether each integer is "almost integer" or not.

Mathematical Background:

The code's logic stems from a mathematical property that states:

2^(n+1) ≈ n!  (mod 10)

or more specifically:

n! / 2^(n+1) ≈ 0.5  (mod 1)

This property implies that the fractional part of the decimal value n! / 2^(n+1) is approximately 0.5. Thus, if the fractional part of this value is close to 0 or 1 (i.e., its last digit is 0 or 9), it indicates that the integer is "almost integer."

import java.math.*;

public class Hickerson {

    final static String LN2 = "0.693147180559945309417232121458";

    public static void main(String[] args) {
        for (int n = 1; n <= 17; n++)
            System.out.printf("%2s is almost integer: %s%n", n, almostInteger(n));
    }

    static boolean almostInteger(int n) {
        BigDecimal a = new BigDecimal(LN2);
        a = a.pow(n + 1).multiply(BigDecimal.valueOf(2));

        long f = n;
        while (--n > 1)
            f *= n;

        BigDecimal b = new BigDecimal(f);
        b = b.divide(a, MathContext.DECIMAL128);

        BigInteger c = b.movePointRight(1).toBigInteger().mod(BigInteger.TEN);

        return c.toString().matches("0|9");
    }
}