Here is a unit test that illustrates the Fly Weight implementation of Integers in Java.

import junit.framework.TestCase;

public final class TestIntegers extends TestCase {
    public void testValueOfGivesTheSameIntegerForSmallValues() {
        assertSame(Integer.valueOf(1), Integer.valueOf(1));
    }

    public void testValueOfGivesADifferentIntegerForSomeValues() {
        assertNotSame(Integer.valueOf(13333333), Integer.valueOf(13333333));
        assertEquals(Integer.valueOf(13333333), Integer.valueOf(13333333));
    }

    public void testNew() {
        assertNotSame(new Integer(1), new Integer(1));
        assertEquals(new Integer(1), new Integer(1));
    }

    public void testValueOfGivesTheSameIntegerForAutoBoxing() {
        assertSame((Integer) 1, (Integer) 1);
    }

    public void testValueOfGivesADifferentIntegerForAutoBoxing() {
        assertNotSame((Integer) 13333333, (Integer) 13333333);
        assertEquals((Integer) 13333333, (Integer) 13333333);
    }
}

Here is why you should use compareTo to find out whether two BigDecimals represent the same number or not.

import java.math.BigDecimal;

import junit.framework.TestCase;

public final class TestBigDecimal extends TestCase {
    public void testBigDecimalZeroEqualsBigDecimalValueOfZero() {
        assertTrue(BigDecimal.ZERO.equals(BigDecimal.valueOf(0)));
    }

    public void testBigDecimalZeroNotEqualsBigDecimalValueOfZeroDot() {
        assertFalse(BigDecimal.ZERO.equals(BigDecimal.valueOf(0.)));
    }

    public void testBigDecimalZeroEqualsBigDecimalValueOfZeroString() {
        assertTrue(BigDecimal.ZERO.equals(new BigDecimal("0")));
    }

    public void testBigDecimalZeroEqualsBigDecimalValueOfZeroDotString() {
        assertTrue(BigDecimal.ZERO.equals(new BigDecimal("0.")));
    }

    public void testBigDecimalZeroNotEqualsBigDecimalValueOfZeroDotZeroString() {
        assertFalse(BigDecimal.ZERO.equals(new BigDecimal("0.0")));
    }

    public void testBigDecimalZeroComparesToBigDecimalValueOfZeroDotAsZero() {
        assertEquals(0, BigDecimal.ZERO.compareTo(BigDecimal.valueOf(0.)));
    }

    public void testBigDecimalZeroComparesToBigDecimalValueOfZeroDotZeroStringAsZero() {
        assertEquals(0, BigDecimal.ZERO.compareTo(new BigDecimal("0.0")));
    }
}