Show that one Gaussian integer divides another

Prove that -1+2i divides 7^16-1 in \mathbb{Z}[i].


First, we claim that 5|(7^16-1). To see this, note that 7^{16}-1 \equiv 2^{16}-1 \equiv (2^4)^4-1 \mod 5. By Fermat’s Little Theorem, 2^4 \equiv 1 \mod 5. Thus 7^{16}-1 \equiv 0 \mod 5 as desired. Note also that -1+2i divides 5 in \mathbb{Z}[i]; in particular, (-1+2i)(-1-2i) = 5. Since divisibility is transitive, -1+2i divides 7^{16}-1.

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