Case 1: The first invertible year after 1800 is 1881.
Case 2: The first invertible year after 1900 is 1961.
Case 3: The first invertible year after 1961 is 6009.
Case 4: The first invertible year after 99 is 101.
Case 5: The first invertible year after 90000 is 90006.
Case 6: The first invertible year after 99999 is 100001.
Case 7: The first invertible year after 1 is 8.
Case 8: The first invertible year after 1000 is 1001.
Case 9: The first invertible year after 1100 is 1111.
Case 10: The first invertible year after 1200 is 1691.
Case 11: The first invertible year after 1300 is 1691.
Case 12: The first invertible year after 1400 is 1691.
Case 13: The first invertible year after 1500 is 1691.
Case 14: The first invertible year after 1600 is 1691.
Case 15: The first invertible year after 1700 is 1881.