X^n-1 is 3-sparse over F_q (q power of 2) if and only if the radical of the odd part m of n divides q^2-1 or m belongs to a specific 7-power family with maximal 7-adic order conditions on q.
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The $3$-sparsity of $X^n-1$ over finite fields, II
X^n-1 is 3-sparse over F_q (q power of 2) if and only if the radical of the odd part m of n divides q^2-1 or m belongs to a specific 7-power family with maximal 7-adic order conditions on q.