The missing years in the Hebrew calendar refer to a chronological discrepancy between Talmudic chronologists for the destruction of the First Temple in 423 BCE (3338 AM) and the modern secular dating for it in 587 BCE.
Both the Babylonian Chronicles and the biblical Chronicles indicate that Nebuchadnezzar captured Jerusalem, but secular scholars have attempted to set a year when the event took place. The Babylonian Chronicles, which were published by Donald Wiseman in 1956, establish that Nebuchadnezzar captured Jerusalem the first time on 2 Adar (16 March) 597 BCE. The Chronicles, with the names of Jewish kings, is derived from British Museum - Cuneiform tablet with part of the Babylonian Chronicle (605-594 BC), (See also Chronicle Concerning the Early Years of Nebuchadnezzar II Reverse, lines 11' - 13')
In the seventh year (of Nebuchadnezzar, 599 BCE) in the month Chislev (Nov/Dec) the king of Babylon assembled his army, and after he had invaded the land of Hatti (northern Syria and southern Anatolia) he laid siege to the city of Judah. On the second day of the month of Adar (16 March) he conquered the city and took the king (Jeconiah) prisoner. He installed in his place a king (Zedekiah) of his own choice, and after he had received rich tribute, he sent forth to Babylon.
2 Chronicles 36:6-10 says:
6 Against him (Jehoiakim) came up Nebuchadnezzar king of Babylon, and bound him in fetters, to carry him to Babylon.
7 Nebuchadnezzar also carried of the vessels of the house of the LORD to Babylon, and put them in his temple at Babylon. 8 ... and Jehoiachin his son reigned in his stead. 9 Jehoiachin was eight years old when he began to reign; and he reigned three months and ten days in Jerusalem...
Before Wiseman's publication, E. R. Thiele had determined from the biblical texts that Nebuchadnezzar's initial capture of Jerusalem occurred in the spring of 597 BCE, while other scholars, including William F. Albright, more frequently dated the event to 598 BCE.