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Amman has never been under Jewish rulership (it was the capital of the Ammonite kingdom). By contrast, Eilat (or neighboring Etzion Gever) was controlled by Shlomo (I Kings 9:26), Yehoshafat (ibid. 22:49), and Uzziah (II Kings 14:22) - it was permanently wrested away only in Achaz's times (ibid. 16:6) - so there is much more reason to consider it part of biblical Israel.