Criteria for 2 to be an e-th power residue of a prime
p ≡ 1 (mod e = ef+1,
have been obtained in various forms for e = 2, 3, 5. Euler proved the well known result that 2 is a quadratic residue of a prime p ≡ ± l (mod 8). Dickson  showed that 2 is a cubic residue of p ≡ 1 (mod 3) if and only if p = L2+27M2 is soluble in integers L, M.
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