In this paper, we settle the last two open cases of non-existence of full quadratic harmonic maps from S 4 to S 5 or S 6 . Assume that there exist full quadratic harmonic maps from S 4 to S n for some integer n. As a consequence of our theorem we obtain that the sufficient and necessary condition of the existence of such maps is that n satisfy 4 ≤ n ≤ 13 and n = 5, 6.
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